Chapter 13
Spectral and Pseudospectral Methods
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After completing this chapter, the reader should be able to:
distinguish spectral, pseudospectral, collocation, tau, and Galerkin formulations;
explain why analytic functions often produce exponential or spectral convergence;
construct Fourier spectral differentiation formulas using Fourier coefficients and FFTs;
construct Chebyshev–Gauss–Lobatto nodes and Chebyshev differentiation matrices;
scale Chebyshev differentiation matrices from [-1,1] to a general interval;
solve linear boundary-value and PDE model problems by Fourier and Chebyshev collocation;
treat nonlinear terms in pseudospectral methods and understand aliasing;
apply dealiasing strategies such as the 2/3-rule and zero padding;
understand filtering, Gibbs oscillations, and stabilization;
compare Fourier, Chebyshev, Legendre, spectral element, and spline-like high-order methods;
use quadrature rules such as Clenshaw–Curtis, Gauss, and Gauss–Lobatto for spectral weak forms;
solve basic, advanced, and research-level exercises on spectral and pseudospectral methods.
Spectral Approximation
A spectral approximation represents a function as a finite expansion u_N(x)=\sum_{k=0}^{N} a_k\phi_k(x), where the basis functions \phi_k are global functions. Common choices include Fourier modes, \phi_k(x)=e^{ikx}, Chebyshev polynomials, \phi_k(x)=T_k(x), and Legendre polynomials, \phi_k(x)=P_k(x).
Finite differences approximate derivatives locally. Spectral methods approximate the entire function globally. If the solution is smooth or analytic, global approximation can converge much faster than any fixed-order local method.
For analytic functions, spectral coefficients typically decay geometrically: |a_k|\le C\rho^{-k}, \qquad \rho>1. Then the truncation error can behave like \|u-u_N\|\le C\rho^{-N}. This is called spectral or exponential convergence.
Spectral, Pseudospectral, Collocation, Tau, and Galerkin Methods
Spectral methods come in several closely related forms.
- Modal spectral method.
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The unknowns are expansion coefficients a_k. Equations are enforced by projecting onto basis functions.
- Nodal pseudospectral method.
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The unknowns are values at nodes u(x_j). Derivatives are computed by differentiating the global interpolant.
- Collocation method.
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The differential equation is enforced pointwise at selected nodes.
- Galerkin spectral method.
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The residual is made orthogonal to the trial space: (R_N,\phi_j)=0.
- Tau method.
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The differential equation is enforced in coefficient space, while some highest modes are replaced by boundary conditions.
Nodal methods are natural for nonlinear terms because products are easy in physical space. Modal methods are natural for analysis because derivatives and projections have clean coefficient formulas. Practical pseudospectral solvers frequently move between the two using fast transforms.
Fourier Series and Periodic Problems
For a 2\pi-periodic function, u(x)=\sum_{k=-\infty}^{\infty}\widehat u_k e^{ikx}, where \widehat u_k=\frac{1}{2\pi}\int_{0}^{2\pi}u(x)e^{-ikx}\,\dd x. The Fourier spectral approximation is u_N(x)=\sum_{|k|\le N/2}\widehat u_k e^{ikx}. Differentiation is diagonal: \frac{\dd}{\dd x}e^{ikx}=ik e^{ikx}. Therefore u_x(x)\approx \sum_{|k|\le N/2} ik\widehat u_k e^{ikx}.
If a periodic function u is analytic in a strip |\operatorname{Im}z|<\sigma, then its Fourier coefficients satisfy an estimate of the form |\widehat u_k|\le C e^{-\sigma |k|}.
For k>0, shift the contour of integration from the real axis to z=x-i\eta, \qquad 0<\eta<\sigma. Analyticity and periodicity allow this shift without changing the integral. Then e^{-ikz}=e^{-ikx}e^{-k\eta}, so |\widehat u_k|\le C_\eta e^{-k\eta}. For k<0, shift to z=x+i\eta. Letting \eta approach the strip width gives the stated exponential decay up to constants.
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If u is analytic and periodic in a strip around the real axis, then the Fourier projection error decays geometrically: \|u-u_N\|_{L^2}\le C e^{-\sigma N}.
By Parseval’s identity, \|u-u_N\|_{L^2}^2 = 2\pi\sum_{|k|>N/2}|\widehat u_k|^2. Using the referenced result, |\widehat u_k|^2\le C^2 e^{-2\sigma |k|}. The tail of a geometric series is geometric. Taking the square root gives the result.
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Discrete Fourier Transform and FFT Differentiation
Let x_j=\frac{2\pi j}{N}, \qquad j=0,\ldots,N-1. The discrete Fourier transform represents nodal values u_j=u(x_j) by coefficients \widehat u_k. A Fourier pseudospectral derivative is computed by: \widehat v_k=ik\widehat u_k, then transforming back: v_j\approx u_x(x_j).
Caption.
Fourier pseudospectral first derivative
periodic nodal vector u=(u_0,\ldots,u_{N-1})
Compute \widehat u=\operatorname{FFT}(u)
Form wavenumbers k=0,1,\ldots,N/2-1,-N/2,\ldots,-1
For each Fourier mode k:
\widehat v_k\gets ik\widehat u_k
v\gets\operatorname{IFFT}(\widehat v)
Return v\approx u_x
The second derivative is obtained by \widehat{u_{xx}}_k=-k^2\widehat u_k.
Fourier Spectral Methods for PDEs
For the periodic heat equation u_t=\kappa u_{xx}, write u_N(x,t)=\sum_k \widehat u_k(t)e^{ikx}. Then \frac{\dd \widehat u_k}{\dd t} = -\kappa k^2\widehat u_k. Thus each mode evolves independently: \widehat u_k(t)=e^{-\kappa k^2 t}\widehat u_k(0).
For periodic advection, u_t+a u_x=0, one obtains \frac{\dd \widehat u_k}{\dd t} = -iak\widehat u_k. Thus \widehat u_k(t)=e^{-iak t}\widehat u_k(0).
For constant-coefficient periodic linear PDEs, Fourier spectral methods diagonalize spatial differential operators. This makes the spatial part exact up to truncation.
Chebyshev Polynomials
Chebyshev polynomials are defined by T_k(x)=\cos(k\arccos x), \qquad -1\le x\le1. The first few are T_0(x)=1, \qquad T_1(x)=x, T_2(x)=2x^2-1, \qquad T_3(x)=4x^3-3x. They satisfy the recurrence T_{k+1}(x)=2xT_k(x)-T_{k-1}(x).
Chebyshev expansions have the form u_N(x)=\sum_{k=0}^{N} a_kT_k(x). They are especially useful on bounded nonperiodic intervals.
Chebyshev–Gauss–Lobatto Nodes
The Chebyshev–Gauss–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. They include both endpoints: x_0=1, \qquad x_N=-1. The nodes cluster near the endpoints.
For N=4, x_0=1, \qquad x_1=\frac{\sqrt2}{2}, \qquad x_2=0, \qquad x_3=-\frac{\sqrt2}{2}, \qquad x_4=-1.
Barycentric Interpolation
Given nodes x_j, the barycentric interpolant is p_N(x)= \frac{\displaystyle\sum_{j=0}^{N} \frac{\lambda_j}{x-x_j}u_j} {\displaystyle\sum_{j=0}^{N} \frac{\lambda_j}{x-x_j}}, where \lambda_j are barycentric weights. For Chebyshev–Lobatto nodes, \lambda_j=(-1)^j\delta_j, with \delta_j= \begin{cases} 1/2, & j=0\text{ or }j=N,\\ 1, & 1\le j\le N-1. \end{cases}
The barycentric formula gives the unique polynomial of degree at most N that interpolates the data (x_j,u_j).
The barycentric formula is algebraically equivalent to the Lagrange interpolation formula p_N(x)=\sum_{j=0}^{N}u_j\ell_j(x), where \ell_j are the cardinal polynomials. Therefore p_N(x_i)=u_i for every node x_i. Uniqueness follows because two polynomials of degree at most N that agree at N+1 distinct points must be identical.
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Chebyshev Differentiation Matrix
Let x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. Define c_j= \begin{cases} 2, & j=0\text{ or }j=N,\\ 1, & 1\le j\le N-1. \end{cases} The Chebyshev first differentiation matrix D=(D_{ij}) is D_{ij} = \frac{c_i}{c_j} \frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j. For interior diagonal entries, D_{ii} = -\frac{x_i}{2(1-x_i^2)}, \qquad 1\le i\le N-1. At the endpoints, D_{00}=\frac{2N^2+1}{6}, \qquad D_{NN}=-\frac{2N^2+1}{6}. The derivative is approximated by u'(x_i)\approx \sum_{j=0}^{N}D_{ij}u(x_j).
Caption.
Chebyshev–Lobatto differentiation matrix
degree N
For j=0,\ldots,N:
x_j\gets\cos(j\pi/N)
c_j\gets2 if j=0 or j=N, otherwise c_j\gets1
For i=0,\ldots,N:
For j=0,\ldots,N:
If i\ne j:
D_{ij}\gets(c_i/c_j)(-1)^{i+j}/(x_i-x_j)
For i=1,\ldots,N-1:
D_{ii}\gets -x_i/[2(1-x_i^2)]
D_{00}\gets(2N^2+1)/6, D_{NN}\gets-(2N^2+1)/6
Return nodes x_j and matrix D
Let p be a polynomial of degree at most N, and let \mathbf p=(p(x_0),\ldots,p(x_N))^T. Then (D\mathbf p)_i=p'(x_i), \qquad i=0,\ldots,N.
The entries of D are derivatives of the Lagrange cardinal functions: D_{ij}=\ell_j'(x_i). Since p has degree at most N, p(x)=\sum_{j=0}^{N}p(x_j)\ell_j(x). Differentiating and evaluating at x_i gives p'(x_i)=\sum_{j=0}^{N}p(x_j)\ell_j'(x_i) = \sum_{j=0}^{N}D_{ij}p(x_j).
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Scaling to General Intervals
For x\in[-1,1], \qquad t\in[a,b], use t=\frac{b-a}{2}x+\frac{a+b}{2}. Then \frac{\dd}{\dd t} = \frac{2}{b-a}\frac{\dd}{\dd x}. Therefore the first derivative matrix on [a,b] is D^{[a,b]}=\frac{2}{b-a}D. The m-th derivative matrix is D_m^{[a,b]}= \left(\frac{2}{b-a}\right)^mD^m.
Chebyshev Collocation for Boundary-Value Problems
Consider -u''(x)=f(x), \qquad -1<x<1, \qquad u(-1)=u(1)=0. Let D^{(2)}=D^2. At interior Chebyshev nodes, -\left(D^{(2)}\mathbf U\right)_i=f(x_i), \qquad i=1,\ldots,N-1. The boundary equations are U_0=u(1)=0, \qquad U_N=u(-1)=0. Equivalently, using the interior index set I=\{1,\ldots,N-1\}, one solves -D^{(2)}_{I,I}\mathbf U_I=\mathbf f_I for homogeneous Dirichlet data.
Chebyshev Collocation for Evolution PDEs
For u_t=\kappa u_{xx}, \qquad -1<x<1, with homogeneous Dirichlet conditions, Chebyshev collocation gives \frac{\dd \mathbf U_I}{\dd t} = \kappa D^{(2)}_{I,I}\mathbf U_I. This is a method-of-lines system. Because Chebyshev nodes cluster near endpoints, the eigenvalues of D^{(2)} can be very large in magnitude, causing severe time step restrictions for explicit schemes.
Chebyshev spatial differentiation matrices can impose very restrictive explicit time steps for parabolic PDEs. Implicit, IMEX, exponential, or stabilized methods are often needed.
Spectral Galerkin Methods
In a spectral Galerkin method, one seeks u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x) such that the residual is orthogonal to the trial space: (R_N,\phi_j)=0, \qquad j=0,\ldots,N. For the Poisson problem with homogeneous boundary conditions, -u''=f, a weak spectral Galerkin formulation is: \int_{-1}^{1}u_N'v_N'\,\dd x = \int_{-1}^{1}fv_N\,\dd x \qquad \forall v_N\in V_N.
Let u solve the weak problem and u_N\in V_N solve the spectral Galerkin problem. Then \int_{-1}^{1}(u-u_N)'v_N'\,\dd x=0 \qquad \forall v_N\in V_N.
The exact solution satisfies \int_{-1}^{1}u'v_N'\,\dd x = \int_{-1}^{1}fv_N\,\dd x for all v_N\in V_N. The Galerkin solution satisfies the same equation with u_N. Subtracting gives the result.
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Tau Methods
Tau methods enforce the differential equation in spectral coefficient form and use some high modes to impose boundary conditions. For example, write u_N(x)=\sum_{k=0}^{N}a_kT_k(x). A tau method may enforce the differential equation for modes k=0,\ldots,N-2, and use the last two equations for u_N(-1)=\alpha, \qquad u_N(1)=\beta. Tau methods are natural in modal spectral computations.
Quadrature in Spectral Methods
Spectral Galerkin methods require integrals such as \int_{-1}^{1}u_N'v_N'\,\dd x, \qquad \int_{-1}^{1}f v_N\,\dd x. Common quadrature rules include: \text{Gauss quadrature}, \qquad \text{Gauss--Lobatto quadrature}, \qquad \text{Clenshaw--Curtis quadrature}. Chebyshev–Lobatto grids naturally pair with Clenshaw–Curtis quadrature.
Clenshaw–Curtis quadrature uses nodes x_j=\cos\left(\frac{j\pi}{N}\right). It integrates the Chebyshev interpolant of the integrand. It is often competitive with Gaussian quadrature for smooth functions and has nested nodes under dyadic refinement.
In linear spectral methods, quadrature errors can affect matrix entries and load vectors. In nonlinear spectral methods, quadrature and interpolation errors can generate aliasing. For nonlinear products, overintegration or dealiasing is often necessary.
Pseudospectral Treatment of Nonlinear Terms
For a nonlinear PDE such as Burgers equation, u_t+u u_x=\nu u_{xx}, a Fourier pseudospectral method usually computes: u_x=\operatorname{IFFT}(ik\widehat u_k), then forms u u_x in physical space, and transforms back. This is efficient because multiplication is cheap in physical space.
However, multiplication of two trigonometric polynomials increases the highest frequency. If u_N(x)=\sum_{|k|\le K}\widehat u_k e^{ikx}, then u_N^2 contains modes up to |k|\le 2K. If only modes |k|\le K are represented, unresolved modes are folded back into lower modes. This is aliasing.
Aliasing
Aliasing occurs when different continuous frequencies are indistinguishable on a finite grid. On an N-point periodic grid, e^{i(k+mN)x_j}=e^{ikx_j}, \qquad x_j=\frac{2\pi j}{N}. Thus modes whose wavenumbers differ by multiples of N have identical nodal values. High frequencies can therefore appear as false low frequencies.
Let x_j=\frac{2\pi j}{N}, \qquad j=0,\ldots,N-1. Then e^{i(k+mN)x_j}=e^{ikx_j} for all integers k,m,j.
Compute e^{i(k+mN)x_j} = e^{ikx_j}e^{imN(2\pi j/N)} = e^{ikx_j}e^{i2\pi mj} = e^{ikx_j}. Since m and j are integers, e^{i2\pi mj}=1.
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Dealiasing: The 2/3-Rule and Zero Padding
For quadratic nonlinearities, the 2/3-rule removes high modes before nonlinear products: \widehat u_k=0 \qquad \text{for} \qquad |k|>\frac{N}{3}. After multiplication, the generated modes remain sufficiently separated to avoid polluting the retained modes.
Another method is zero padding: \widehat u \longrightarrow \widehat u_{\mathrm{pad}}, transforming to a larger physical grid, multiplying there, transforming back, and truncating.
Caption.
Fourier pseudospectral nonlinear term with 2/3-dealiasing
Fourier coefficients \widehat u_k, grid size N
Set \widehat u_k=0 for |k|>N/3
Compute u=\operatorname{IFFT}(\widehat u)
Compute \widehat{u_x}_k=ik\widehat u_k
Compute u_x=\operatorname{IFFT}(\widehat{u_x})
Form nonlinear product w=u\,u_x in physical space
Compute \widehat w=\operatorname{FFT}(w)
Set \widehat w_k=0 for |k|>N/3
Return dealiased nonlinear coefficients \widehat w
Filtering
Spectral filters damp high modes: \widehat u_k^{\,\mathrm{filtered}} = \sigma\left(\frac{|k|}{K}\right)\widehat u_k. A common exponential filter is \sigma(\eta)=\exp(-\alpha \eta^p), \qquad 0\le\eta\le1, where p is even and \alpha>0.
For Chebyshev expansions, u_N(x)=\sum_{k=0}^{N}a_kT_k(x), one filters coefficients by a_k^{\mathrm{filtered}} = \sigma(k/N)a_k.
Gibbs Phenomenon
If a function has a jump discontinuity, its Fourier partial sums exhibit oscillatory overshoots near the discontinuity. This is the Gibbs phenomenon. The overshoot does not disappear in amplitude as N\to\infty, although it becomes localized near the jump.
Filtering, Gegenbauer reconstruction, shock-capturing viscosity, and adaptive domain decomposition can reduce the visible oscillations.
Fourier Pseudospectral Burgers Solver
For viscous Burgers equation u_t+u u_x=\nu u_{xx}, the Fourier pseudospectral semidiscretization is \frac{\dd \widehat u_k}{\dd t} = -\widehat{u u_x}_k - \nu k^2\widehat u_k. A practical implementation computes the nonlinear term in physical space, applies dealiasing, and advances in time using RK4, IMEX, exponential time differencing, or semi-implicit schemes.
Caption.
One RK4 step for Fourier pseudospectral Burgers equation
coefficients \widehat u^n, viscosity \nu, time step \Delta t
Define F(\widehat u)=-\widehat{u u_x}-\nu k^2\widehat u, with dealiasing in the product
K_1\gets F(\widehat u^n)
K_2\gets F(\widehat u^n+\Delta t K_1/2)
K_3\gets F(\widehat u^n+\Delta t K_2/2)
K_4\gets F(\widehat u^n+\Delta t K_3)
\widehat u^{n+1}\gets \widehat u^n+\frac{\Delta t}{6}(K_1+2K_2+2K_3+K_4)
Return \widehat u^{n+1}
Chebyshev Pseudospectral Nonlinear BVPs
For a nonlinear BVP -u''+\gamma u^3=f, \qquad u(-1)=u(1)=0, Chebyshev collocation gives interior residuals R_i(U)= -\left(D^2U\right)_i+\gamma U_i^3-f(x_i), \qquad i=1,\ldots,N-1. Boundary residuals are R_0(U)=U_0, \qquad R_N(U)=U_N. Newton’s method solves J(U^{(m)})\Delta U=-R(U^{(m)}), with Jacobian entries J_{ij}=-D^{(2)}_{ij}+3\gamma U_i^2\delta_{ij} for interior rows, and boundary rows enforcing U_0=U_N=0.
Spectral Accuracy and Loss of Spectral Accuracy
For analytic functions, spectral methods may converge exponentially. For functions with only s derivatives, convergence is algebraic. A typical statement is \|u-u_N\|\le C N^{-s}\|u\|_{H^s}. For discontinuous functions, Gibbs oscillations appear, and convergence in maximum norm may fail near jumps.
- Analytic functions.
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Geometric or exponential coefficient decay.
- Infinitely smooth but nonanalytic functions.
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Faster than algebraic in many cases, but not necessarily exponential.
- Finite smoothness.
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Algebraic convergence depending on the number of derivatives.
- Discontinuities.
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Slow convergence, Gibbs oscillations, and the need for filtering or shock-capturing ideas.
Legendre Spectral Methods
Legendre polynomials P_k(x) are orthogonal on [-1,1] with weight one: \int_{-1}^{1}P_j(x)P_k(x)\,\dd x = \frac{2}{2k+1}\delta_{jk}. Legendre methods are popular for spectral element and Galerkin formulations. They often pair naturally with Gauss or Gauss–Lobatto quadrature.
A Legendre–Galerkin approximation takes u_N(x)=\sum_{k=0}^{N}a_kP_k(x) or uses basis combinations that satisfy boundary conditions.
Ultraspherical Spectral Methods
Ultraspherical methods represent derivatives in related polynomial bases. Instead of forming dense, ill-conditioned differentiation matrices, they use sparse conversion and differentiation operators between ultraspherical polynomial spaces. This can lead to well-conditioned spectral discretizations for differential equations.
Spectral Element Methods
Spectral element methods divide the domain into elements and use high-degree polynomials on each element. They combine: \text{geometric flexibility of finite elements} with \text{high-order accuracy of spectral methods}. A typical nodal spectral element method uses Gauss–Lobatto nodes on each element. Mass matrices may be diagonal or nearly diagonal when Gauss–Lobatto quadrature is used.
Comparison with Splines and High-Order Finite Elements
Spectral methods use global high-degree polynomials or global Fourier modes. High-order finite elements and spline methods use piecewise polynomials. Splines and isogeometric methods provide high continuity and geometric flexibility.
Global spectral methods are extremely efficient for smooth solutions on simple domains. Spectral elements, high-order finite elements, and splines are better for complex geometry, local refinement, discontinuities, and multiphysics coupling.
Practical Comparison of Spectral Tools
- Fourier spectral.
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Best for smooth periodic problems. Fast, accurate, and diagonal for constant coefficient derivatives.
- Chebyshev collocation.
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Best for smooth nonperiodic interval problems. Simple differentiation matrices and excellent accuracy.
- Legendre Galerkin.
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Natural for weak forms and spectral element methods.
- Tau method.
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Efficient in modal coefficient formulations with boundary conditions.
- Pseudospectral nonlinear methods.
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Efficient for nonlinear PDEs, but require dealiasing and filtering.
- Spectral element methods.
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High-order accuracy with geometric flexibility and elementwise locality.
- Ultraspherical methods.
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Useful for well-conditioned spectral discretizations of differential equations.
- Filtering.
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Helpful for underresolved solutions, nonlinear aliasing, and Gibbs oscillations, but must be used carefully to avoid damaging accuracy.
Exercises
The following exercise bank is intentionally large. Basic problems test definitions, formulas, and direct computations. Starred exercises require derivations, stability analysis, aliasing analysis, or spectral approximation theory. Problems marked \star, \star\star, and \star\star\star are progressively harder.
Basic problems
Write the general spectral approximation u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). Give two possible choices of \phi_k.
Explain the difference between a spectral and a pseudospectral method.
If u(x)=\sum_k\widehat u_k e^{ikx}, write the Fourier coefficients of u_x and u_{xx}.
For u_t=\kappa u_{xx}, derive the ODE satisfied by each Fourier coefficient.
Compute T_0(x),\quad T_1(x),\quad T_2(x),\quad T_3(x).
Write the Chebyshev–Lobatto nodes for N=4.
Explain why Chebyshev–Lobatto nodes cluster near the endpoints.
State the off-diagonal formula for the Chebyshev differentiation matrix.
State the diagonal entries of the Chebyshev differentiation matrix.
Derive the scaling D^{[a,b]}=\frac{2}{b-a}D.
Explain how to form a Chebyshev second derivative matrix.
Write the interior Chebyshev collocation equations for -u''=f,\qquad u(-1)=u(1)=0.
What does it mean for a spectral Galerkin residual to be orthogonal to the trial space?
Explain the basic idea of a tau method.
What nodes are used in Clenshaw–Curtis quadrature?
Define aliasing on a finite periodic grid.
Show that e^{i(k+N)x_j}=e^{ikx_j} on the grid x_j=2\pi j/N.
State the 2/3-dealiasing rule.
Give one example of a spectral filter.
What is the Gibbs phenomenon?
Write the Fourier pseudospectral semidiscretization of viscous Burgers equation.
Write the Chebyshev collocation residual for -u''+u^3=f.
State the orthogonality relation for Legendre polynomials.
Explain the basic idea of a spectral element method.
Give one situation where Fourier methods are preferable and one situation where Chebyshev methods are preferable.
Intermediate problems \star
Prove the referenced result.
Prove the referenced result.
Derive the FFT-based Fourier pseudospectral differentiation algorithm.
Solve u_t+a u_x=0 in Fourier coefficient form.
Prove T_{k+1}(x)=2xT_k(x)-T_{k-1}(x).
Prove the referenced result.
Prove the referenced result.
Derive D_{00}=\frac{2N^2+1}{6}, \qquad D_{NN}=-\frac{2N^2+1}{6}.
Derive the mapping from [-1,1] to [a,b] and the derivative scaling.
Construct the method-of-lines system for u_t=u_{xx},\qquad u(-1,t)=u(1,t)=0.
Prove the referenced result.
Write a Chebyshev tau method for -u''+u=f,\qquad u(-1)=u(1)=0.
Explain why Clenshaw–Curtis quadrature is natural on Chebyshev–Lobatto nodes.
Prove the referenced result.
Show that the product of two trigonometric polynomials with maximum wavenumber K contains modes up to 2K.
Explain why the 2/3-rule is used for quadratic nonlinearities.
Describe zero-padding dealiasing for a quadratic pseudospectral product.
Analyze the filter \sigma(\eta)=\exp(-\alpha\eta^p). What happens as p increases?
Explain why Gibbs oscillations become localized but do not vanish in overshoot amplitude.
Derive the right-hand side for a Fourier pseudospectral Burgers solver.
Derive the Newton Jacobian for R_i(U)=-(D^2U)_i+\gamma U_i^3-f_i.
Formulate a Legendre Galerkin method for -u''=f,\qquad u(-1)=u(1)=0.
Explain why Gauss–Lobatto quadrature can produce diagonal mass matrices in nodal spectral element methods.
Compare global spectral methods with high-order finite elements.
Explain how excessive filtering can reduce spectral accuracy.
Advanced problems \star\star
Prove exponential decay of Chebyshev coefficients for functions analytic in a Bernstein ellipse.
Derive an interpolation error estimate for Chebyshev–Lobatto interpolation.
Derive all entries of the Chebyshev differentiation matrix from Lagrange cardinal functions.
Derive the recurrence for differentiating a Chebyshev series in coefficient space.
Analyze the conditioning of Chebyshev collocation matrices for second-order BVPs.
Compare Chebyshev differentiation-matrix methods with spectral integration methods.
Analyze RK4 time-stepping stability for a Fourier spectral discretization of the heat equation.
Derive an exponential time-differencing method for a Fourier spectral discretization of a semilinear PDE.
Show that multiplication in physical space corresponds to convolution in Fourier space.
Give a detailed proof of the 2/3-dealiasing rule for quadratic products.
Determine a safe truncation rule for cubic nonlinearities.
Study phase-shift dealiasing and compare it with zero padding.
Design a spectral viscosity method for a nonlinear conservation law.
Explain how Gegenbauer reconstruction can recover high-order accuracy away from discontinuities.
Design a filter that preserves modes k\le N/2 almost exactly and damps only the highest modes.
Derive derivative relations for Legendre polynomial expansions.
Determine the Gauss quadrature order needed to exactly integrate products of polynomials in a Legendre Galerkin method.
Analyze how variable coefficients affect sparsity and accuracy in spectral Galerkin methods.
Derive the basic ultraspherical spectral discretization for a second-order linear ODE.
Derive local stiffness and mass matrices for a one-dimensional spectral element.
Study how nonmatching spectral element grids can be coupled using mortar methods.
Use a coordinate map to cluster Chebyshev points near a boundary layer and analyze the effect on differentiation matrices.
Study rational Chebyshev methods for problems on semi-infinite intervals.
Design a Fourier pseudospectral splitting method for the nonlinear Schrodinger equation.
Design a Fourier pseudospectral method for the KdV equation and discuss stiffness from the third derivative.
Research-level problems \star\star\star
Prove a rigorous spectral convergence theorem for Chebyshev collocation applied to a linear elliptic boundary-value problem.
Study convergence of a dealiased Fourier pseudospectral method for a nonlinear parabolic PDE.
Develop a Fourier pseudospectral projection method for incompressible Navier–Stokes equations on a periodic box.
Analyze the role of dealiasing in turbulent pseudospectral simulations.
Investigate how spectral filters affect discrete energy estimates.
Study entropy-stable spectral or spectral element discretizations for conservation laws.
Develop a spectral viscosity or filtering strategy for shock capturing in a pseudospectral solver.
Design an hp-adaptive spectral element method for a solution with localized singularities.
Study fast Chebyshev transforms and their relation to FFTs.
Develop a matrix-free Chebyshev collocation solver for a high-resolution BVP.
Design preconditioners for spectral discretizations of elliptic problems.
Study spherical harmonic spectral methods for PDEs on the sphere.
Compare mapped spectral methods, spectral elements, and high-order finite elements on complex domains.
Develop a spectral method for a fractional Laplacian problem and compare Fourier and Galerkin formulations.
Design a spectral collocation method for a Caputo time-fractional differential equation.
Connect spectral collocation in time with spectral deferred correction methods.
Formulate a space-time spectral method for a parabolic PDE and analyze its nonlinear algebraic system.
Investigate interval or radii-polynomial certification for spectral approximations of nonlinear boundary-value problems.
Compare spectral residual minimization with physics-informed neural networks for a smooth nonlinear PDE.
Design reproducible benchmarks comparing Fourier spectral, Chebyshev collocation, tau, Galerkin, spectral element, and filtered pseudospectral methods.
Exercise solutions
The solutions below are converted from the uploaded LaTeX solution file. Open an accordion to view the full solution.
This chapter gives detailed worked solutions for the exercises in Chapter 13. Each solution includes the problem formulation, the method, the mathematical derivation, the conclusion, and a diagnostic comment.
Exercise 13.1
Problem formulation.
Write the general spectral approximation u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). Give two possible choices of \phi_k.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.2
Problem formulation.
Explain the difference between a spectral and a pseudospectral method.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.3
Problem formulation.
If u(x)=\sum_k\widehat u_k e^{ikx}, write the Fourier coefficients of u_x and u_{xx}.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.4
Problem formulation.
For u_t=\kappa u_{xx}, derive the ODE satisfied by each Fourier coefficient.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.5
Problem formulation.
Compute T_0(x),\quad T_1(x),\quad T_2(x),\quad T_3(x).
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.6
Problem formulation.
Write the Chebyshev–Lobatto nodes for N=4.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.7
Problem formulation.
Explain why Chebyshev–Lobatto nodes cluster near the endpoints.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.8
Problem formulation.
State the off-diagonal formula for the Chebyshev differentiation matrix.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.9
Problem formulation.
State the diagonal entries of the Chebyshev differentiation matrix.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.10
Problem formulation.
Derive the scaling D^{[a,b]}=\frac{2}{b-a}D.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.11
Problem formulation.
Explain how to form a Chebyshev second derivative matrix.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.12
Problem formulation.
Write the interior Chebyshev collocation equations for -u''=f,\qquad u(-1)=u(1)=0.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.13
Problem formulation.
What does it mean for a spectral Galerkin residual to be orthogonal to the trial space?
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.14
Problem formulation.
Explain the basic idea of a tau method.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.15
Problem formulation.
What nodes are used in Clenshaw–Curtis quadrature?
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.16
Problem formulation.
Define aliasing on a finite periodic grid.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Aliasing occurs because discrete samples cannot distinguish Fourier modes whose indices differ by multiples of the number of grid points. For N grid points, e^{i(k+mN)x_j}=e^{ikx_j}. In nonlinear products, high modes generated by convolution fold back into low modes. The 2/3-rule removes modes with |k|>\frac{N}{3} before multiplication, preventing quadratic nonlinearities from contaminating the resolved band.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.17
Problem formulation.
Show that e^{i(k+N)x_j}=e^{ikx_j} on the grid x_j=2\pi j/N.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Aliasing occurs because discrete samples cannot distinguish Fourier modes whose indices differ by multiples of the number of grid points. For N grid points, e^{i(k+mN)x_j}=e^{ikx_j}. In nonlinear products, high modes generated by convolution fold back into low modes. The 2/3-rule removes modes with |k|>\frac{N}{3} before multiplication, preventing quadratic nonlinearities from contaminating the resolved band.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.18
Problem formulation.
State the 2/3-dealiasing rule.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Aliasing occurs because discrete samples cannot distinguish Fourier modes whose indices differ by multiples of the number of grid points. For N grid points, e^{i(k+mN)x_j}=e^{ikx_j}. In nonlinear products, high modes generated by convolution fold back into low modes. The 2/3-rule removes modes with |k|>\frac{N}{3} before multiplication, preventing quadratic nonlinearities from contaminating the resolved band.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.19
Problem formulation.
Give one example of a spectral filter.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.20
Problem formulation.
What is the Gibbs phenomenon?
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.21
Problem formulation.
Write the Fourier pseudospectral semidiscretization of viscous Burgers equation.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.22
Problem formulation.
Write the Chebyshev collocation residual for -u''+u^3=f.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.23
Problem formulation.
State the orthogonality relation for Legendre polynomials.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.24
Problem formulation.
Explain the basic idea of a spectral element method.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.25
Problem formulation.
Give one situation where Fourier methods are preferable and one situation where Chebyshev methods are preferable.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.26
Problem formulation.
Prove the referenced result.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.27
Problem formulation.
Prove the referenced result.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 13.28
Problem formulation.
Derive the FFT-based Fourier pseudospectral differentiation algorithm.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.29
Problem formulation.
Solve u_t+a u_x=0 in Fourier coefficient form.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.30
Problem formulation.
Prove T_{k+1}(x)=2xT_k(x)-T_{k-1}(x).
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.31
Problem formulation.
Prove the referenced result.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.32
Problem formulation.
Prove the referenced result.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.33
Problem formulation.
Derive D_{00}=\frac{2N^2+1}{6}, \qquad D_{NN}=-\frac{2N^2+1}{6}.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.34
Problem formulation.
Derive the mapping from [-1,1] to [a,b] and the derivative scaling.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.35
Problem formulation.
Construct the method-of-lines system for u_t=u_{xx},\qquad u(-1,t)=u(1,t)=0.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.36
Problem formulation.
Prove the referenced result.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.37
Problem formulation.
Write a Chebyshev tau method for -u''+u=f,\qquad u(-1)=u(1)=0.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.38
Problem formulation.
Explain why Clenshaw–Curtis quadrature is natural on Chebyshev–Lobatto nodes.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.39
Problem formulation.
Prove the referenced result.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Aliasing occurs because discrete samples cannot distinguish Fourier modes whose indices differ by multiples of the number of grid points. For N grid points, e^{i(k+mN)x_j}=e^{ikx_j}. In nonlinear products, high modes generated by convolution fold back into low modes. The 2/3-rule removes modes with |k|>\frac{N}{3} before multiplication, preventing quadratic nonlinearities from contaminating the resolved band.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.40
Problem formulation.
Show that the product of two trigonometric polynomials with maximum wavenumber K contains modes up to 2K.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Aliasing occurs because discrete samples cannot distinguish Fourier modes whose indices differ by multiples of the number of grid points. For N grid points, e^{i(k+mN)x_j}=e^{ikx_j}. In nonlinear products, high modes generated by convolution fold back into low modes. The 2/3-rule removes modes with |k|>\frac{N}{3} before multiplication, preventing quadratic nonlinearities from contaminating the resolved band.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.41
Problem formulation.
Explain why the 2/3-rule is used for quadratic nonlinearities.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.42
Problem formulation.
Describe zero-padding dealiasing for a quadratic pseudospectral product.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Aliasing occurs because discrete samples cannot distinguish Fourier modes whose indices differ by multiples of the number of grid points. For N grid points, e^{i(k+mN)x_j}=e^{ikx_j}. In nonlinear products, high modes generated by convolution fold back into low modes. The 2/3-rule removes modes with |k|>\frac{N}{3} before multiplication, preventing quadratic nonlinearities from contaminating the resolved band.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.43
Problem formulation.
Analyze the filter \sigma(\eta)=\exp(-\alpha\eta^p). What happens as p increases?
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.44
Problem formulation.
Explain why Gibbs oscillations become localized but do not vanish in overshoot amplitude.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.45
Problem formulation.
Derive the right-hand side for a Fourier pseudospectral Burgers solver.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.46
Problem formulation.
Derive the Newton Jacobian for R_i(U)=-(D^2U)_i+\gamma U_i^3-f_i.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Verify the stationarity residual, feasibility residual, and descent or curvature condition, not only the final numerical value.
Exercise 13.47
Problem formulation.
Formulate a Legendre Galerkin method for -u''=f,\qquad u(-1)=u(1)=0.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.48
Problem formulation.
Explain why Gauss–Lobatto quadrature can produce diagonal mass matrices in nodal spectral element methods.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.49
Problem formulation.
Compare global spectral methods with high-order finite elements.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 13.50
Problem formulation.
Explain how excessive filtering can reduce spectral accuracy.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.51
Problem formulation.
Prove exponential decay of Chebyshev coefficients for functions analytic in a Bernstein ellipse.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.52
Problem formulation.
Derive an interpolation error estimate for Chebyshev–Lobatto interpolation.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 13.53
Problem formulation.
Derive all entries of the Chebyshev differentiation matrix from Lagrange cardinal functions.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.54
Problem formulation.
Derive the recurrence for differentiating a Chebyshev series in coefficient space.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.55
Problem formulation.
Analyze the conditioning of Chebyshev collocation matrices for second-order BVPs.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 13.56
Problem formulation.
Compare Chebyshev differentiation-matrix methods with spectral integration methods.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.57
Problem formulation.
Analyze RK4 time-stepping stability for a Fourier spectral discretization of the heat equation.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Always test the result on the scalar model equation or Fourier mode and verify the amplification factor satisfies the stated stability bound.
Exercise 13.58
Problem formulation.
Derive an exponential time-differencing method for a Fourier spectral discretization of a semilinear PDE.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.59
Problem formulation.
Show that multiplication in physical space corresponds to convolution in Fourier space.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.60
Problem formulation.
Give a detailed proof of the 2/3-dealiasing rule for quadratic products.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Aliasing occurs because discrete samples cannot distinguish Fourier modes whose indices differ by multiples of the number of grid points. For N grid points, e^{i(k+mN)x_j}=e^{ikx_j}. In nonlinear products, high modes generated by convolution fold back into low modes. The 2/3-rule removes modes with |k|>\frac{N}{3} before multiplication, preventing quadratic nonlinearities from contaminating the resolved band.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.61
Problem formulation.
Determine a safe truncation rule for cubic nonlinearities.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Aliasing occurs because discrete samples cannot distinguish Fourier modes whose indices differ by multiples of the number of grid points. For N grid points, e^{i(k+mN)x_j}=e^{ikx_j}. In nonlinear products, high modes generated by convolution fold back into low modes. The 2/3-rule removes modes with |k|>\frac{N}{3} before multiplication, preventing quadratic nonlinearities from contaminating the resolved band.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 13.62
Problem formulation.
Study phase-shift dealiasing and compare it with zero padding.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Aliasing occurs because discrete samples cannot distinguish Fourier modes whose indices differ by multiples of the number of grid points. For N grid points, e^{i(k+mN)x_j}=e^{ikx_j}. In nonlinear products, high modes generated by convolution fold back into low modes. The 2/3-rule removes modes with |k|>\frac{N}{3} before multiplication, preventing quadratic nonlinearities from contaminating the resolved band.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.63
Problem formulation.
Design a spectral viscosity method for a nonlinear conservation law.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.64
Problem formulation.
Explain how Gegenbauer reconstruction can recover high-order accuracy away from discontinuities.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 13.65
Problem formulation.
Design a filter that preserves modes k\le N/2 almost exactly and damps only the highest modes.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.66
Problem formulation.
Derive derivative relations for Legendre polynomial expansions.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.67
Problem formulation.
Determine the Gauss quadrature order needed to exactly integrate products of polynomials in a Legendre Galerkin method.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 13.68
Problem formulation.
Analyze how variable coefficients affect sparsity and accuracy in spectral Galerkin methods.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.69
Problem formulation.
Derive the basic ultraspherical spectral discretization for a second-order linear ODE.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 13.70
Problem formulation.
Derive local stiffness and mass matrices for a one-dimensional spectral element.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Always test the result on the scalar model equation or Fourier mode and verify the amplification factor satisfies the stated stability bound.
Exercise 13.71
Problem formulation.
Study how nonmatching spectral element grids can be coupled using mortar methods.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.72
Problem formulation.
Use a coordinate map to cluster Chebyshev points near a boundary layer and analyze the effect on differentiation matrices.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.73
Problem formulation.
Study rational Chebyshev methods for problems on semi-infinite intervals.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.74
Problem formulation.
Design a Fourier pseudospectral splitting method for the nonlinear Schrodinger equation.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.75
Problem formulation.
Design a Fourier pseudospectral method for the KdV equation and discuss stiffness from the third derivative.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Always test the result on the scalar model equation or Fourier mode and verify the amplification factor satisfies the stated stability bound.
Exercise 13.76
Problem formulation.
Prove a rigorous spectral convergence theorem for Chebyshev collocation applied to a linear elliptic boundary-value problem.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 13.77
Problem formulation.
Study convergence of a dealiased Fourier pseudospectral method for a nonlinear parabolic PDE.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 13.78
Problem formulation.
Develop a Fourier pseudospectral projection method for incompressible Navier–Stokes equations on a periodic box.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.79
Problem formulation.
Analyze the role of dealiasing in turbulent pseudospectral simulations.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Aliasing occurs because discrete samples cannot distinguish Fourier modes whose indices differ by multiples of the number of grid points. For N grid points, e^{i(k+mN)x_j}=e^{ikx_j}. In nonlinear products, high modes generated by convolution fold back into low modes. The 2/3-rule removes modes with |k|>\frac{N}{3} before multiplication, preventing quadratic nonlinearities from contaminating the resolved band.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.80
Problem formulation.
Investigate how spectral filters affect discrete energy estimates.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Always test the result on the scalar model equation or Fourier mode and verify the amplification factor satisfies the stated stability bound.
Exercise 13.81
Problem formulation.
Study entropy-stable spectral or spectral element discretizations for conservation laws.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.82
Problem formulation.
Develop a spectral viscosity or filtering strategy for shock capturing in a pseudospectral solver.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.83
Problem formulation.
Design an hp-adaptive spectral element method for a solution with localized singularities.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.84
Problem formulation.
Study fast Chebyshev transforms and their relation to FFTs.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.85
Problem formulation.
Develop a matrix-free Chebyshev collocation solver for a high-resolution BVP.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Chebyshev polynomials satisfy T_k(x)=\cos(k\arccos x). The Chebyshev–Lobatto nodes are x_j=\cos\left(\frac{j\pi}{N}\right), \qquad j=0,\ldots,N. The differentiation matrix has off-diagonal entries D_{ij}=\frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, \qquad i\ne j, where c_0=c_N=2 and c_j=1 otherwise. The diagonal entries are chosen so that each row sums to zero: D_{ii}=-\sum_{j\ne i}D_{ij}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.86
Problem formulation.
Design preconditioners for spectral discretizations of elliptic problems.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Distinguish forward error from backward error; a small residual does not imply a small forward error when the problem is ill-conditioned.
Exercise 13.87
Problem formulation.
Study spherical harmonic spectral methods for PDEs on the sphere.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.88
Problem formulation.
Compare mapped spectral methods, spectral elements, and high-order finite elements on complex domains.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 13.89
Problem formulation.
Develop a spectral method for a fractional Laplacian problem and compare Fourier and Galerkin formulations.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 13.90
Problem formulation.
Design a spectral collocation method for a Caputo time-fractional differential equation.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 13.91
Problem formulation.
Connect spectral collocation in time with spectral deferred correction methods.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.92
Problem formulation.
Formulate a space-time spectral method for a parabolic PDE and analyze its nonlinear algebraic system.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.93
Problem formulation.
Investigate interval or radii-polynomial certification for spectral approximations of nonlinear boundary-value problems.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.94
Problem formulation.
Compare spectral residual minimization with physics-informed neural networks for a smooth nonlinear PDE.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
Spectral methods approximate the solution by global basis functions: u_N(x)=\sum_{k=0}^{N}a_k\phi_k(x). In collocation methods, the differential equation is enforced at nodes: \mathcal L u_N(x_j)=f(x_j). In Galerkin methods, the residual is made orthogonal to the test space: \langle \mathcal L u_N-f,\phi_j\rangle=0. For smooth or analytic solutions, the coefficients a_k decay rapidly, leading to spectral accuracy.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 13.95
Problem formulation.
Design reproducible benchmarks comparing Fourier spectral, Chebyshev collocation, tau, Galerkin, spectral element, and filtered pseudospectral methods.
Method.
Use Fourier/Chebyshev expansions, spectral differentiation matrices, aliasing identities, filtering, collocation, and Galerkin/tau formulations.
Detailed solution and justification.
For a periodic function, u(x)=\sum_{k=-K}^{K}\widehat u_k e^{ikx}. Differentiating term by term, u_x(x)=\sum_{k=-K}^{K}ik\widehat u_k e^{ikx}. Thus in Fourier space, \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k. This is the basis of Fourier spectral differentiation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Interactive tools
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