Chapter 12
Numerical Methods for Partial Differential Equations
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After completing this chapter, the reader should be able to:
identify elliptic, parabolic, hyperbolic, and nonlinear PDE model problems;
derive finite-difference schemes for heat, wave, advection, and Poisson equations;
analyze stability by von Neumann analysis and CFL restrictions;
formulate method-of-lines semidiscretizations;
derive implicit, Crank–Nicolson, ADI, and IMEX time-stepping methods;
construct weak forms and Galerkin finite-element discretizations for PDEs;
evaluate PDE Galerkin integrals using high-order quadrature rules;
treat nonlinear PDEs by Picard, Newton, Newton–Krylov, and continuation methods;
understand finite-volume conservation, numerical fluxes, and shock capturing;
derive Fourier spectral and Chebyshev spectral differentiation formulas;
understand spline, B-spline, and isogeometric PDE discretizations;
compare finite difference, finite volume, finite element, spline, and spectral methods in practical applications.
Model Partial Differential Equations
The canonical model equations are: u_t=\kappa u_{xx} \qquad \text{heat equation}, u_{tt}=c^2u_{xx} \qquad \text{wave equation}, -\Delta u=f \qquad \text{Poisson equation}, and u_t+a u_x=0 \qquad \text{linear advection equation}. Nonlinear PDEs include: u_t+u u_x=\nu u_{xx} \qquad \text{viscous Burgers equation}, u_t=\Delta u+F(u) \qquad \text{reaction--diffusion equation}, and i u_t+\Delta u+\sigma |u|^2u=0 \qquad \text{nonlinear Schrodinger equation}.
Numerical PDE methods combine three ingredients: spatial discretization, time discretization, and algebraic solution strategy. For nonlinear PDEs, the algebraic systems are usually nonlinear and require Newton, Picard, fixed-point, or Newton–Krylov iterations.
Classification and Numerical Consequences
Elliptic equations such as -\Delta u=f typically lead to large sparse linear or nonlinear algebraic systems.
Parabolic equations such as u_t=\kappa \Delta u combine diffusion smoothing with time evolution. Explicit schemes face severe time-step restrictions: \Delta t=O(\Delta x^2).
Hyperbolic equations such as u_t+a u_x=0 transport information along characteristics and require CFL conditions of the type \frac{|a|\Delta t}{\Delta x}\le C.
Nonlinear conservation laws can develop shocks even from smooth initial data, so stability, monotonicity, entropy conditions, and limiting become essential.
Finite Differences for the Heat Equation
Consider u_t=\kappa u_{xx}, \qquad 0<x<L. Let x_i=i\Delta x, \qquad t^n=n\Delta t, and write U_i^n\approx u(x_i,t^n). Using forward Euler in time and a central second difference in space gives \frac{U_i^{n+1}-U_i^n}{\Delta t} = \kappa \frac{U_{i-1}^{n}-2U_i^n+U_{i+1}^n}{\Delta x^2}. Therefore U_i^{n+1} = U_i^n+\mu \left( U_{i-1}^n-2U_i^n+U_{i+1}^n \right), \qquad \mu=\frac{\kappa\Delta t}{\Delta x^2}.
Caption.
Explicit heat-equation scheme
initial data U_i^0, boundary values, \mu=\kappa\Delta t/\Delta x^2
For n=0,1,2,\ldots:
For interior nodes i=1,\ldots,N-1:
U_i^{n+1}\gets U_i^n+\mu(U_{i-1}^n-2U_i^n+U_{i+1}^n)
Apply boundary conditions
For the one-dimensional heat equation, the explicit scheme U_i^{n+1} = U_i^n+\mu(U_{i-1}^n-2U_i^n+U_{i+1}^n) is von Neumann stable if and only if 0\le\mu\le\frac12.
Insert the Fourier mode U_i^n=G^n e^{i\theta i}. Then G = 1+\mu(e^{-i\theta}-2+e^{i\theta}) = 1-4\mu\sin^2\left(\frac{\theta}{2}\right). Stability requires |G|\le1 for every \theta. Since G is real, this is equivalent to -1\le 1-4\mu\sin^2(\theta/2)\le 1. The most restrictive case is \sin^2(\theta/2)=1, which gives \mu\le\frac12. Also \mu\ge0 is required for diffusion.
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Implicit and Crank–Nicolson Heat Schemes
Backward Euler in time gives \frac{U_i^{n+1}-U_i^n}{\Delta t} = \kappa \frac{U_{i-1}^{n+1}-2U_i^{n+1}+U_{i+1}^{n+1}}{\Delta x^2}. Thus -\mu U_{i-1}^{n+1} +(1+2\mu)U_i^{n+1} -\mu U_{i+1}^{n+1} = U_i^n. This requires solving a tridiagonal system at every step.
Crank–Nicolson averages the spatial operator at times n and n+1: \frac{U_i^{n+1}-U_i^n}{\Delta t} = \frac{\kappa}{2} \left[ \frac{U_{i-1}^{n+1}-2U_i^{n+1}+U_{i+1}^{n+1}}{\Delta x^2} + \frac{U_{i-1}^{n}-2U_i^{n}+U_{i+1}^{n}}{\Delta x^2} \right].
For sufficiently smooth solutions of the heat equation, the Crank–Nicolson method is second order in time and second order in space: \text{error}=O(\Delta t^2+\Delta x^2).
The time discretization is the trapezoidal rule applied to the semidiscrete heat equation, hence second order in time. The centered second-difference approximation to u_{xx} is second order in space. Combining the two local consistency errors gives O(\Delta t^2+\Delta x^2). Stability of the method for the heat equation transfers consistency to convergence.
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Finite Differences for the Wave Equation
For u_{tt}=c^2u_{xx}, use the centered scheme \frac{U_i^{n+1}-2U_i^n+U_i^{n-1}}{\Delta t^2} = c^2 \frac{U_{i-1}^{n}-2U_i^n+U_{i+1}^{n}}{\Delta x^2}. Thus U_i^{n+1} = 2U_i^n-U_i^{n-1} + \lambda^2 \left( U_{i-1}^{n}-2U_i^n+U_{i+1}^{n} \right), \qquad \lambda=\frac{c\Delta t}{\Delta x}.
The centered scheme for the one-dimensional wave equation is stable under the CFL condition \lambda=\frac{c\Delta t}{\Delta x}\le1.
Insert U_i^n=G^n e^{i\theta i}. The amplification equation is G+G^{-1}-2 = -4\lambda^2\sin^2(\theta/2). Equivalently, G+G^{-1} = 2\left[1-2\lambda^2\sin^2(\theta/2)\right]. For roots on the unit circle, the right-hand side must lie in [-2,2]. The most restrictive case is \sin^2(\theta/2)=1, which yields \lambda\le1.
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Advection: Upwind, Lax–Friedrichs, and Lax–Wendroff
For u_t+a u_x=0, with a>0, the upwind scheme is U_i^{n+1} = U_i^n-\nu(U_i^n-U_{i-1}^n), \qquad \nu=\frac{a\Delta t}{\Delta x}. For a<0, the stencil is reversed.
The Lax–Friedrichs scheme is U_i^{n+1} = \frac12(U_{i+1}^n+U_{i-1}^n) - \frac{\nu}{2}(U_{i+1}^n-U_{i-1}^n). The Lax–Wendroff scheme is U_i^{n+1} = U_i^n - \frac{\nu}{2}(U_{i+1}^n-U_{i-1}^n) + \frac{\nu^2}{2} (U_{i+1}^n-2U_i^n+U_{i-1}^n).
Upwind schemes are robust and dissipative. Lax–Friedrichs is simple but strongly diffusive. Lax–Wendroff is second order but can create oscillations near discontinuities. Nonlinear conservation laws require flux limiters, TVD schemes, ENO, WENO, or Godunov-type methods.
Method of Lines
The method of lines discretizes space first and keeps time continuous. For the heat equation, u_t=\kappa u_{xx}, a finite-difference semidiscretization gives \frac{\dd U}{\dd t} = \kappa A_h U, ] where \(A_h\) is the discrete Laplacian. One then applies ODE solvers: \[ U'(t)=F(t,U(t)). For nonlinear PDEs, u_t=\mathcal N(u) becomes U'(t)=N_h(U).
Caption.
Method of lines for a parabolic PDE
spatial grid, spatial operator L_h, nonlinear term N_h, initial data U(0)
Form semidiscrete ODE system U'=L_hU+N_h(U)
Choose time integrator: explicit RK, implicit RK, BDF, IMEX, or exponential method
Advance U(t) in time with error control
Apply boundary conditions at each step or in the operator
Poisson and Laplace Equations
For -\Delta u=f on a rectangular domain, the standard five-point finite-difference stencil is -\Delta_h U_{i,j} = \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2} = f_{i,j}. This gives a sparse linear system AU=F.
If u\in C^4, then the five-point Laplacian satisfies \Delta_h u(x_i,y_j) = \Delta u(x_i,y_j)+O(h^2).
Apply the one-dimensional centered second-difference expansion in the x-direction and in the y-direction: \frac{u(x+h,y)-2u(x,y)+u(x-h,y)}{h^2} = u_{xx}(x,y)+O(h^2), \frac{u(x,y+h)-2u(x,y)+u(x,y-h)}{h^2} = u_{yy}(x,y)+O(h^2). Adding the two formulas gives the result.
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Finite Element Weak Forms for PDEs
For the Poisson problem -\Delta u=f \qquad \text{in }\Omega, \qquad u=0 \qquad \text{on }\partial\Omega, multiply by a test function v\in H_0^1(\Omega) and integrate: \int_\Omega -\Delta u\,v\,\dd x = \int_\Omega fv\,\dd x. Integration by parts gives \int_\Omega \nabla u\cdot\nabla v\,\dd x = \int_\Omega fv\,\dd x. The Galerkin method is: \text{Find }u_h\in V_h\subset H_0^1(\Omega) \text{ such that } \int_\Omega \nabla u_h\cdot\nabla v_h\,\dd x = \int_\Omega f v_h\,\dd x \quad \forall v_h\in V_h.
Let u solve the weak Poisson problem and u_h solve the finite-element Galerkin problem. Then \int_\Omega \nabla(u-u_h)\cdot\nabla v_h\,\dd x=0 \qquad \forall v_h\in V_h.
The exact weak solution satisfies \int_\Omega\nabla u\cdot\nabla v_h\,\dd x = \int_\Omega f v_h\,\dd x \qquad \forall v_h\in V_h. The Galerkin solution satisfies the same equation with u_h in place of u. Subtracting gives the orthogonality identity.
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Quadrature for PDE Galerkin Integrals
Finite-element PDE methods require integrals over elements: K_{ij}^{(K)} = \int_K \nabla\phi_j\cdot\nabla\phi_i\,\dd x, M_{ij}^{(K)} = \int_K \phi_j\phi_i\,\dd x, F_i^{(K)} = \int_K f\phi_i\,\dd x. On a reference element \widehat K, with map x=F_K(\widehat x), one uses \int_K g(x)\,\dd x = \int_{\widehat K} g(F_K(\widehat x)) |\det J_K(\widehat x)|\,\dd \widehat x.
For tensor-product elements, a tensor-product Gauss rule is \int_{-1}^{1}\int_{-1}^{1} G(\xi,\eta)\,\dd\xi\,\dd\eta \approx \sum_{r=1}^{m}\sum_{s=1}^{m} w_r w_s G(\xi_r,\xi_s). For triangles, one uses symmetric triangular quadrature rules.
If the basis functions have degree k, then: \phi_i\phi_j \quad \text{has degree }2k, while \nabla\phi_i\cdot\nabla\phi_j \quad \text{has degree }2k-2. Variable coefficients and nonlinear terms increase the degree or destroy polynomial exactness. Use high-order quadrature and check convergence by increasing the quadrature order.
Nonlinear PDEs and Newton Linearization
Consider the nonlinear elliptic PDE -\Delta u+\gamma u^3=f \qquad \text{in }\Omega, \qquad u=0 \qquad \text{on }\partial\Omega. The weak residual is R(u;v) = \int_\Omega \nabla u\cdot\nabla v\,\dd x + \int_\Omega \gamma u^3v\,\dd x - \int_\Omega fv\,\dd x. The nonlinear Galerkin method seeks u_h\in V_h \quad \text{such that} \quad R(u_h;v_h)=0 \quad \forall v_h\in V_h.
Newton’s method solves for a correction \delta_h: R'(u_h^{(k)})[\delta_h;v_h] = -R(u_h^{(k)};v_h), where R'(u)[w;v] = \int_\Omega \nabla w\cdot\nabla v\,\dd x + \int_\Omega 3\gamma u^2wv\,\dd x.
Caption.
Newton–Galerkin method for a nonlinear elliptic PDE
finite-element space V_h, initial guess u_h^{(0)}
For k=0,1,2,\ldots:
Assemble residual vector
\item R_i=R(u_h^{(k)};\phi_i) \item
Assemble Jacobian matrix
\item J_{ij}=R'(u_h^{(k)})[\phi_j;\phi_i] \item
Solve J\Delta U=-R
U^{(k+1)}\gets U^{(k)}+\Delta U
If \|R\| and \|\Delta U\| are small enough:
Return u_h^{(k+1)}
Picard, Newton–Krylov, and Continuation for Nonlinear PDEs
Picard iteration freezes nonlinear coefficients. For example, -\Delta u+\gamma u^3=f can be iterated as -\Delta u^{(k+1)}+\gamma (u^{(k)})^2u^{(k+1)}=f. Picard is often more robust but slower than Newton.
Newton–Krylov methods solve the Newton linear system approximately: J(U^{(k)})\Delta U=-R(U^{(k)}) using Krylov methods such as GMRES. Matrix-free approximations use J(U)v \approx \frac{R(U+\epsilon v)-R(U)}{\epsilon}. Continuation introduces a parameter: -\Delta u+\lambda u^3=f, ] and increases \(\lambda\) gradually. \section{Reaction--Diffusion Equations} A reaction--diffusion equation has the form \[ u_t=D\Delta u+F(u). A method-of-lines discretization gives U'(t)=D A_h U+F_h(U). If the diffusion is stiff but reaction is nonlinear, an IMEX scheme treats diffusion implicitly and reaction explicitly: \frac{U^{n+1}-U^n}{\Delta t} = D A_h U^{n+1}+F_h(U^n). A fully implicit method solves U^{n+1}-\Delta t D A_hU^{n+1} - \Delta t F_h(U^{n+1}) = U^n.
Burgers Equation and Nonlinear Conservation Laws
The viscous Burgers equation is u_t+\left(\frac{u^2}{2}\right)_x=\nu u_{xx}. The inviscid case u_t+\left(\frac{u^2}{2}\right)_x=0 can form shocks. Conservative finite-volume schemes update cell averages: \overline U_j^{n+1} = \overline U_j^n - \frac{\Delta t}{\Delta x} \left( F_{j+1/2}^n-F_{j-1/2}^n \right). Here F_{j+1/2} is a numerical flux.
For periodic boundary conditions, the finite-volume update \overline U_j^{n+1} = \overline U_j^n - \frac{\Delta t}{\Delta x} \left( F_{j+1/2}^n-F_{j-1/2}^n \right) conserves the total discrete mass: \sum_j \overline U_j^{n+1} = \sum_j \overline U_j^n.
Sum the update over j: \sum_j \overline U_j^{n+1} = \sum_j \overline U_j^n - \frac{\Delta t}{\Delta x} \sum_j \left( F_{j+1/2}^n-F_{j-1/2}^n \right). The flux sum telescopes. Under periodic boundary conditions, the boundary fluxes cancel, so the total mass is preserved.
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Finite-Volume Numerical Fluxes
For a scalar conservation law u_t+f(u)_x=0, common numerical fluxes include the Lax–Friedrichs flux F_{j+1/2} = \frac12\left[f(U_j)+f(U_{j+1})\right] - \frac{\alpha}{2}(U_{j+1}-U_j), where \alpha\ge \max |f'(u)|. The Godunov flux solves the local Riemann problem exactly or approximately.
High-resolution methods use reconstruction: U_{j+1/2}^{-}, \qquad U_{j+1/2}^{+}, and apply a numerical flux F_{j+1/2}=\mathcal F(U_{j+1/2}^{-},U_{j+1/2}^{+}). Limiters, ENO, and WENO reconstructions reduce spurious oscillations near shocks.
Fourier Spectral Methods
For periodic problems on x\in[0,2\pi], approximate u(x,t) \approx \sum_{k=-N/2}^{N/2-1} \widehat u_k(t)e^{ikx}. Then \frac{\partial u}{\partial x} \approx \sum_k ik\widehat u_k e^{ikx}, and \frac{\partial^2 u}{\partial x^2} \approx \sum_k -k^2\widehat u_k e^{ikx}. Thus differentiation is diagonal in Fourier space: \widehat{u_x}_k=ik\widehat u_k, \qquad \widehat{u_{xx}}_k=-k^2\widehat u_k.
For the heat equation, u_t=\kappa u_{xx}, the Fourier coefficients satisfy \frac{\dd \widehat u_k}{\dd t} = -\kappa k^2\widehat u_k. Therefore \widehat u_k(t)=e^{-\kappa k^2t}\widehat u_k(0).
Caption.
Fourier pseudospectral differentiation
periodic values u_j=u(x_j), Fourier wavenumbers k
Compute Fourier coefficients \widehat u=\operatorname{FFT}(u)
Compute \widehat v_k=ik\widehat u_k
Compute v=\operatorname{IFFT}(\widehat v)
Return v\approx u_x
If a periodic function is analytic in a strip around the real axis, then Fourier spectral differentiation converges geometrically with respect to the number of modes, up to roundoff limits.
Analytic periodic functions have Fourier coefficients that decay geometrically: |\widehat u_k|\le C e^{-\sigma |k|}. Differentiation multiplies the k-th coefficient by ik, which is only algebraic growth. Algebraic growth is dominated by geometric decay. Hence the truncated differentiated Fourier series converges geometrically.
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Aliasing and the Dealiasing Rule
For nonlinear PDEs, products such as u u_x are computed in physical space and transformed back to Fourier space. Multiplication creates modes beyond the truncation range. These unresolved modes fold back into the resolved range, causing aliasing.
The standard 2/3-dealiasing rule sets high modes to zero: \widehat u_k=0 \qquad \text{for } |k|>\frac{N}{3}. This reduces aliasing errors for quadratic nonlinearities.
Chebyshev Spectral Methods
For nonperiodic intervals, Chebyshev methods use x_j=\cos\left(\frac{\pi j}{N}\right), \qquad j=0,\ldots,N. Let D be the Chebyshev differentiation matrix. Then u_x(\mathbf x)\approx D\mathbf u, \qquad u_{xx}(\mathbf x)\approx D^2\mathbf u. For -u''=f, \qquad u(-1)=u(1)=0, one solves the interior system -D_{II}^{(2)}U_I=f_I, \qquad I=\{1,\ldots,N-1\}.
Spline and B-Spline Methods
Spline methods approximate the solution using piecewise polynomials with smoothness across elements. A B-spline basis of degree p is denoted B_{i,p}(x). A spline approximation is u_h(x)=\sum_i U_i B_{i,p}(x). Unlike standard C^0 finite elements, splines can have higher continuity: C^{p-1} for simple knots. This is useful for PDEs involving higher derivatives, such as beam, plate, and fourth-order problems.
For a Galerkin spline method for Poisson, \int_\Omega \nabla u_h\cdot\nabla v_h\,\dd x = \int_\Omega f v_h\,\dd x. The matrices are assembled as K_{ij} = \int_\Omega \nabla B_j\cdot\nabla B_i\,\dd x, \qquad F_i=\int_\Omega fB_i\,\dd x.
B-splines and NURBS provide smooth basis functions and exact geometry representation in computer-aided design. Isogeometric analysis uses these bases for PDE discretization. The price is more complex quadrature, connectivity, and implementation.
Spline Collocation
Spline collocation chooses u_h(x)=\sum_i U_iB_{i,p}(x) and enforces the PDE at collocation points: \mathcal L u_h(x_j)=f(x_j). For nonlinear PDEs, \mathcal N(u_h)(x_j)=0. The resulting nonlinear algebraic system can be solved by Newton’s method. Spline collocation can achieve high accuracy with smooth approximants.
Discontinuous Galerkin Methods
Discontinuous Galerkin methods allow trial functions to be discontinuous across element interfaces. For hyperbolic conservation laws, DG methods combine local polynomial approximation with numerical fluxes. In one dimension, \int_{K_j} u_{h,t}v_h\,\dd x - \int_{K_j} f(u_h)v_h'\,\dd x + \widehat f_{j+1/2}v_h(x_{j+1}^{-}) - \widehat f_{j-1/2}v_h(x_j^{+}) = 0. The numerical flux couples neighboring elements.
DG methods are local, high-order, and well suited for conservation laws, adaptive meshes, and complex geometries.
Operator Splitting and IMEX Methods
Many PDEs have the form u_t=A(u)+B(u), where A may be stiff and B may be nonlinear. Lie splitting uses e^{\Delta t(A+B)} \approx e^{\Delta t A}e^{\Delta t B}. Strang splitting uses e^{\Delta t(A+B)} \approx e^{\frac{\Delta t}{2}A} e^{\Delta t B} e^{\frac{\Delta t}{2}A}. IMEX methods treat the stiff part implicitly and the nonstiff part explicitly: \frac{U^{n+1}-U^n}{\Delta t} = A U^{n+1}+B(U^n).
ADI Methods
For multidimensional parabolic equations, u_t=\kappa(u_{xx}+u_{yy}), implicit time stepping leads to large sparse systems. Alternating-direction implicit methods split the solve into one-dimensional implicit solves.
A Peaceman–Rachford ADI step is \left(I-\frac{\Delta t}{2}A_x\right)U^{n+1/2} = \left(I+\frac{\Delta t}{2}A_y\right)U^n, \left(I-\frac{\Delta t}{2}A_y\right)U^{n+1} = \left(I+\frac{\Delta t}{2}A_x\right)U^{n+1/2}.
Solvers and Preconditioners for PDE Systems
Discretized PDEs lead to large algebraic systems: AU=b or R(U)=0. Important linear solvers include: \text{CG}, \qquad \text{GMRES}, \qquad \text{multigrid}, \qquad \text{domain decomposition}. For nonlinear PDEs, Newton–Krylov methods solve J(U^k)\Delta U=-R(U^k) approximately. Good preconditioners are often the difference between success and failure.
Practical Comparison of PDE Methods
- Finite differences.
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Simple, efficient, and excellent on structured grids. Less flexible for complex geometries.
- Finite volumes.
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Conservative by construction. Essential for conservation laws, shocks, and fluid dynamics.
- Finite elements.
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Flexible for geometry, boundary conditions, adaptivity, and weak formulations.
- Discontinuous Galerkin.
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High-order, conservative, local, and well suited to hyperbolic problems and adaptivity.
- Fourier spectral.
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Extremely accurate for smooth periodic problems. FFT-based and diagonal for constant coefficient derivatives.
- Chebyshev spectral.
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Extremely accurate for smooth nonperiodic interval problems. Dense matrices and endpoint clustering require care.
- Spline and isogeometric methods.
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Smooth high-order bases and exact CAD geometry. Excellent for higher-order PDEs and geometric applications.
- Nonlinear PDE solvers.
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Need robust Newton, Picard, continuation, damping, preconditioning, and accurate quadrature for nonlinear residuals.
- Quadrature.
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Low-order quadrature can corrupt high-order PDE methods. Always match quadrature order to basis degree, coefficient complexity, and nonlinear terms.
Exercises
The following exercise bank is intentionally large. Basic problems test definitions, computations, and essential formulas. Starred exercises require proofs, stability analysis, weak formulations, nonlinear linearization, quadrature design, or spectral approximation arguments. Problems marked \star, \star\star, and \star\star\star are progressively harder.
Basic problems
Write one elliptic, one parabolic, and one hyperbolic model PDE.
Derive the explicit finite-difference scheme for u_t=\kappa u_{xx}.
State the stability condition for the explicit heat scheme in one space dimension.
Write the Crank–Nicolson scheme for the heat equation.
Derive the centered finite-difference scheme for u_{tt}=c^2u_{xx}.
State the CFL condition for the centered wave scheme.
Write the upwind scheme for u_t+a u_x=0 when a>0.
Write the Lax–Friedrichs scheme for linear advection.
Write the Lax–Wendroff scheme for linear advection.
Explain the method of lines.
Write the five-point stencil for -\Delta u=f.
Derive the weak form of the homogeneous Dirichlet Poisson problem.
Write the finite-element stiffness matrix entry for Poisson’s equation.
Explain why quadrature is needed in finite-element assembly.
Write the weak residual for -\Delta u+u^3=f.
Derive the Newton Jacobian for -\Delta u+\gamma u^3=f.
Write the finite-volume update for a scalar conservation law.
If u(x)=\sum_k \widehat u_k e^{ikx}, write the Fourier coefficients of u_x.
Explain how a Chebyshev differentiation matrix approximates u_x.
What is a B-spline basis and why can it be useful for PDEs?
Explain the basic idea of discontinuous Galerkin methods.
Explain why IMEX methods are useful for PDEs with stiff and nonstiff terms.
Explain the idea of alternating-direction implicit methods.
What is aliasing in Fourier pseudospectral methods?
Give one advantage of finite differences, finite volumes, finite elements, and spectral methods.
Intermediate problems \star
Prove the referenced result.
Prove the referenced result.
Prove the referenced result.
Use von Neumann analysis to show the CFL condition for the upwind scheme.
Derive the Lax–Wendroff scheme using Taylor expansion in time.
Derive the modified equation of the upwind scheme.
Write the semidiscrete matrix system for the heat equation with Dirichlet boundary conditions.
Prove the referenced result.
Construct the two-dimensional Poisson matrix on a 3\times3 interior grid.
Prove the referenced result.
For quadratic finite elements and polynomial coefficient a(x) of degree 2, choose a Gauss quadrature order for exact stiffness integration.
Derive the Newton Jacobian for R(u;v)=\int_\Omega\nabla u\cdot\nabla v+\int_\Omega u^3v-\int_\Omega fv.
Derive a Picard iteration for -\Delta u+\gamma u^3=f.
Explain the finite-difference Jacobian-vector product used in matrix-free Newton–Krylov methods.
Derive an IMEX scheme for u_t=D\Delta u+F(u).
Prove the referenced result.
Derive the Lax–Friedrichs numerical flux for a scalar conservation law.
Explain the Godunov flux for Burgers equation.
Solve the heat equation in Fourier space.
Prove the referenced result.
Explain the 2/3-dealiasing rule for quadratic nonlinearities.
Write the Chebyshev collocation system for -u''=f,\qquad u(-1)=u(1)=0.
Write the stiffness matrix entries for a B-spline Galerkin discretization of Poisson’s equation.
Derive the local DG weak form for a scalar conservation law.
Derive the Peaceman–Rachford ADI scheme for the two-dimensional heat equation.
Advanced problems \star\star
State and prove the Lax equivalence theorem in a simplified linear setting.
Use an energy method to prove stability of an implicit heat-equation scheme.
Prove a discrete maximum principle for the explicit heat scheme under \mu\le1/2.
Analyze numerical dispersion for the centered wave-equation scheme.
Derive the artificial viscosity term in the modified equation of Lax–Friedrichs.
Explain total variation diminishing schemes and derive a flux-limiter form.
Derive the basic idea of WENO reconstruction for a scalar conservation law.
Explain entropy solutions and entropy-stable numerical fluxes.
Prove an H^1-error estimate for linear finite elements applied to Poisson’s equation.
Use the Aubin–Nitsche duality argument to obtain an L^2-error estimate.
Analyze the effect of underintegration on finite-element consistency.
State local convergence conditions for Newton’s method applied to nonlinear PDE discretizations.
Design a continuation method for a nonlinear elliptic PDE with parameter \lambda.
Derive a stable time-stepping method for u_t=\epsilon^2\Delta u+u-u^3.
Derive a Strang splitting method for the nonlinear Schrodinger equation.
Formulate a Fourier pseudospectral method for viscous Burgers equation and include dealiasing.
Investigate conditioning of Chebyshev discretizations for elliptic PDEs.
Design a spectral filter for stabilizing a pseudospectral nonlinear PDE solver.
Derive the Cox–de Boor recursion for B-splines.
Derive the stiffness matrix entries for an isogeometric Poisson discretization.
Explain how knot multiplicity affects B-spline smoothness.
Analyze stability of an upwind DG method for linear advection.
Derive a symmetric interior penalty DG method for Poisson’s equation.
Design a multigrid V-cycle for the finite-difference Poisson equation.
Explain additive Schwarz preconditioning for PDE systems.
Research-level problems \star\star\star
Study full approximation scheme multigrid for nonlinear elliptic PDEs.
Design physics-based preconditioners for Newton–Krylov discretizations of nonlinear PDEs.
Develop residual estimators for adaptive finite elements applied to nonlinear elliptic PDEs.
Derive a dual-weighted residual estimator for a quantity of interest.
Study entropy-stable finite-volume or DG schemes for systems of conservation laws.
Develop a fifth-order WENO method and analyze its behavior near smooth extrema.
Design a limiter or artificial-viscosity strategy for DG shock capturing.
Discuss Fourier pseudospectral methods for incompressible turbulence and the role of dealiasing.
Derive a projection method for the incompressible Navier–Stokes equations.
Design energy-stable schemes for Cahn–Hilliard or Allen–Cahn equations.
Study numerical methods preserving mass, energy, positivity, or symplectic structure for PDEs.
Formulate a space-time Galerkin method for a parabolic PDE.
Develop a spectral element method combining high-order polynomial elements and Gauss–Lobatto quadrature.
Design an hp-adaptive finite-element method for elliptic PDEs with singularities.
Study isogeometric discretization and Newton linearization for nonlinear PDEs.
Compare finite difference, spectral, and Galerkin methods for fractional Laplacian PDEs.
Develop a POD–Galerkin reduced-order model for a nonlinear parabolic PDE.
Compare Galerkin finite elements with residual-minimizing neural solvers for nonlinear PDEs.
Study stochastic Galerkin or stochastic collocation methods for PDEs with random coefficients.
Design reproducible benchmarks comparing finite differences, finite volumes, finite elements, splines, Fourier spectral, Chebyshev spectral, and DG methods.
Exercise solutions
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This chapter gives detailed worked solutions for the exercises in Chapter 12. Each solution includes the problem formulation, the method, the mathematical derivation, the conclusion, and a diagnostic comment.
Exercise 12.1
Problem formulation.
Write one elliptic, one parabolic, and one hyperbolic model PDE.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.2
Problem formulation.
Derive the explicit finite-difference scheme for u_t=\kappa u_{xx}.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For u_t=\kappa u_{xx}, use forward Euler in time and centered differences in space: \frac{U_i^{n+1}-U_i^n}{\Delta t} = \kappa\frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. Thus U_i^{n+1}=U_i^n+\mu(U_{i-1}^n-2U_i^n+U_{i+1}^n), where \mu=\frac{\kappa\Delta t}{\Delta x^2}. Von Neumann analysis with U_i^n=\xi^ne^{i\theta i} gives \xi=1-4\mu\sin^2(\theta/2). Stability requires |\xi|\le1, so in one dimension \mu\le\frac12.
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 12.3
Problem formulation.
State the stability condition for the explicit heat scheme in one space dimension.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For u_t=\kappa u_{xx}, use forward Euler in time and centered differences in space: \frac{U_i^{n+1}-U_i^n}{\Delta t} = \kappa\frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. Thus U_i^{n+1}=U_i^n+\mu(U_{i-1}^n-2U_i^n+U_{i+1}^n), where \mu=\frac{\kappa\Delta t}{\Delta x^2}. Von Neumann analysis with U_i^n=\xi^ne^{i\theta i} gives \xi=1-4\mu\sin^2(\theta/2). Stability requires |\xi|\le1, so in one dimension \mu\le\frac12.
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 12.4
Problem formulation.
Write the Crank–Nicolson scheme for the heat equation.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For u_t=\kappa u_{xx}, use forward Euler in time and centered differences in space: \frac{U_i^{n+1}-U_i^n}{\Delta t} = \kappa\frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. Thus U_i^{n+1}=U_i^n+\mu(U_{i-1}^n-2U_i^n+U_{i+1}^n), where \mu=\frac{\kappa\Delta t}{\Delta x^2}. Von Neumann analysis with U_i^n=\xi^ne^{i\theta i} gives \xi=1-4\mu\sin^2(\theta/2). Stability requires |\xi|\le1, so in one dimension \mu\le\frac12.
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 12.5
Problem formulation.
Derive the centered finite-difference scheme for u_{tt}=c^2u_{xx}.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For u_{tt}=c^2u_{xx}, use centered differences: \frac{U_i^{n+1}-2U_i^n+U_i^{n-1}}{\Delta t^2} = c^2\frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. Thus U_i^{n+1} = 2U_i^n-U_i^{n-1} + \lambda^2(U_{i-1}^n-2U_i^n+U_{i+1}^n), where \lambda=\frac{c\Delta t}{\Delta x}. Von Neumann analysis gives the CFL condition \lambda\le1.
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 12.6
Problem formulation.
State the CFL condition for the centered wave scheme.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For u_{tt}=c^2u_{xx}, use centered differences: \frac{U_i^{n+1}-2U_i^n+U_i^{n-1}}{\Delta t^2} = c^2\frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. Thus U_i^{n+1} = 2U_i^n-U_i^{n-1} + \lambda^2(U_{i-1}^n-2U_i^n+U_{i+1}^n), where \lambda=\frac{c\Delta t}{\Delta x}. Von Neumann analysis gives the CFL condition \lambda\le1.
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 12.7
Problem formulation.
Write the upwind scheme for u_t+a u_x=0 when a>0.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.8
Problem formulation.
Write the Lax–Friedrichs scheme for linear advection.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.9
Problem formulation.
Write the Lax–Wendroff scheme for linear advection.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.10
Problem formulation.
Explain the method of lines.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.11
Problem formulation.
Write the five-point stencil for -\Delta u=f.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For Poisson’s equation -\Delta u=f on a square grid, the five-point discretization is -\Delta_hU_{i,j} = \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2}. The discrete equation is \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2} = f_{i,j}. The truncation error is O(h^2) for smooth u.
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 12.12
Problem formulation.
Derive the weak form of the homogeneous Dirichlet Poisson problem.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For Poisson’s equation -\Delta u=f on a square grid, the five-point discretization is -\Delta_hU_{i,j} = \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2}. The discrete equation is \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2} = f_{i,j}. The truncation error is O(h^2) for smooth u.
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 12.13
Problem formulation.
Write the finite-element stiffness matrix entry for Poisson’s equation.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For Poisson’s equation -\Delta u=f on a square grid, the five-point discretization is -\Delta_hU_{i,j} = \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2}. The discrete equation is \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2} = f_{i,j}. The truncation error is O(h^2) for smooth u.
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 12.14
Problem formulation.
Explain why quadrature is needed in finite-element assembly.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.15
Problem formulation.
Write the weak residual for -\Delta u+u^3=f.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.16
Problem formulation.
Derive the Newton Jacobian for -\Delta u+\gamma u^3=f.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.17
Problem formulation.
Write the finite-volume update for a scalar conservation law.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.18
Problem formulation.
If u(x)=\sum_k \widehat u_k e^{ikx}, write the Fourier coefficients of u_x.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.19
Problem formulation.
Explain how a Chebyshev differentiation matrix approximates u_x.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.20
Problem formulation.
What is a B-spline basis and why can it be useful for PDEs?
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.21
Problem formulation.
Explain the basic idea of discontinuous Galerkin methods.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.22
Problem formulation.
Explain why IMEX methods are useful for PDEs with stiff and nonstiff terms.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.23
Problem formulation.
Explain the idea of alternating-direction implicit methods.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.24
Problem formulation.
What is aliasing in Fourier pseudospectral methods?
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.25
Problem formulation.
Give one advantage of finite differences, finite volumes, finite elements, and spectral methods.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.26
Problem formulation.
Prove the referenced result.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For u_t=\kappa u_{xx}, use forward Euler in time and centered differences in space: \frac{U_i^{n+1}-U_i^n}{\Delta t} = \kappa\frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. Thus U_i^{n+1}=U_i^n+\mu(U_{i-1}^n-2U_i^n+U_{i+1}^n), where \mu=\frac{\kappa\Delta t}{\Delta x^2}. Von Neumann analysis with U_i^n=\xi^ne^{i\theta i} gives \xi=1-4\mu\sin^2(\theta/2). Stability requires |\xi|\le1, so in one dimension \mu\le\frac12.
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 12.27
Problem formulation.
Prove the referenced result.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.28
Problem formulation.
Prove the referenced result.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For u_{tt}=c^2u_{xx}, use centered differences: \frac{U_i^{n+1}-2U_i^n+U_i^{n-1}}{\Delta t^2} = c^2\frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. Thus U_i^{n+1} = 2U_i^n-U_i^{n-1} + \lambda^2(U_{i-1}^n-2U_i^n+U_{i+1}^n), where \lambda=\frac{c\Delta t}{\Delta x}. Von Neumann analysis gives the CFL condition \lambda\le1.
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 12.29
Problem formulation.
Use von Neumann analysis to show the CFL condition for the upwind scheme.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.30
Problem formulation.
Derive the Lax–Wendroff scheme using Taylor expansion in time.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.31
Problem formulation.
Derive the modified equation of the upwind scheme.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.32
Problem formulation.
Write the semidiscrete matrix system for the heat equation with Dirichlet boundary conditions.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For u_t=\kappa u_{xx}, use forward Euler in time and centered differences in space: \frac{U_i^{n+1}-U_i^n}{\Delta t} = \kappa\frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. Thus U_i^{n+1}=U_i^n+\mu(U_{i-1}^n-2U_i^n+U_{i+1}^n), where \mu=\frac{\kappa\Delta t}{\Delta x^2}. Von Neumann analysis with U_i^n=\xi^ne^{i\theta i} gives \xi=1-4\mu\sin^2(\theta/2). Stability requires |\xi|\le1, so in one dimension \mu\le\frac12.
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 12.33
Problem formulation.
Prove the referenced result.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For Poisson’s equation -\Delta u=f on a square grid, the five-point discretization is -\Delta_hU_{i,j} = \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2}. The discrete equation is \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2} = f_{i,j}. The truncation error is O(h^2) for smooth u.
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 12.34
Problem formulation.
Construct the two-dimensional Poisson matrix on a 3\times3 interior grid.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For Poisson’s equation -\Delta u=f on a square grid, the five-point discretization is -\Delta_hU_{i,j} = \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2}. The discrete equation is \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2} = f_{i,j}. The truncation error is O(h^2) for smooth u.
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 12.35
Problem formulation.
Prove the referenced result.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.36
Problem formulation.
For quadratic finite elements and polynomial coefficient a(x) of degree 2, choose a Gauss quadrature order for exact stiffness integration.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.37
Problem formulation.
Derive the Newton Jacobian for R(u;v)=\int_\Omega\nabla u\cdot\nabla v+\int_\Omega u^3v-\int_\Omega fv.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.38
Problem formulation.
Derive a Picard iteration for -\Delta u+\gamma u^3=f.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.39
Problem formulation.
Explain the finite-difference Jacobian-vector product used in matrix-free Newton–Krylov methods.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.40
Problem formulation.
Derive an IMEX scheme for u_t=D\Delta u+F(u).
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.41
Problem formulation.
Prove the referenced result.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.42
Problem formulation.
Derive the Lax–Friedrichs numerical flux for a scalar conservation law.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.43
Problem formulation.
Explain the Godunov flux for Burgers equation.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.44
Problem formulation.
Solve the heat equation in Fourier space.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For u_t=\kappa u_{xx}, use forward Euler in time and centered differences in space: \frac{U_i^{n+1}-U_i^n}{\Delta t} = \kappa\frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. Thus U_i^{n+1}=U_i^n+\mu(U_{i-1}^n-2U_i^n+U_{i+1}^n), where \mu=\frac{\kappa\Delta t}{\Delta x^2}. Von Neumann analysis with U_i^n=\xi^ne^{i\theta i} gives \xi=1-4\mu\sin^2(\theta/2). Stability requires |\xi|\le1, so in one dimension \mu\le\frac12.
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 12.45
Problem formulation.
Prove the referenced result.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.46
Problem formulation.
Explain the 2/3-dealiasing rule for quadratic nonlinearities.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.47
Problem formulation.
Write the Chebyshev collocation system for -u''=f,\qquad u(-1)=u(1)=0.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For Poisson’s equation -\Delta u=f on a square grid, the five-point discretization is -\Delta_hU_{i,j} = \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2}. The discrete equation is \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2} = f_{i,j}. The truncation error is O(h^2) for smooth u.
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 12.48
Problem formulation.
Write the stiffness matrix entries for a B-spline Galerkin discretization of Poisson’s equation.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For Poisson’s equation -\Delta u=f on a square grid, the five-point discretization is -\Delta_hU_{i,j} = \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2}. The discrete equation is \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2} = f_{i,j}. The truncation error is O(h^2) for smooth u.
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 12.49
Problem formulation.
Derive the local DG weak form for a scalar conservation law.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.50
Problem formulation.
Derive the Peaceman–Rachford ADI scheme for the two-dimensional heat equation.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For u_t=\kappa u_{xx}, use forward Euler in time and centered differences in space: \frac{U_i^{n+1}-U_i^n}{\Delta t} = \kappa\frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. Thus U_i^{n+1}=U_i^n+\mu(U_{i-1}^n-2U_i^n+U_{i+1}^n), where \mu=\frac{\kappa\Delta t}{\Delta x^2}. Von Neumann analysis with U_i^n=\xi^ne^{i\theta i} gives \xi=1-4\mu\sin^2(\theta/2). Stability requires |\xi|\le1, so in one dimension \mu\le\frac12.
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 12.51
Problem formulation.
State and prove the Lax equivalence theorem in a simplified linear setting.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.52
Problem formulation.
Use an energy method to prove stability of an implicit heat-equation scheme.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For u_t=\kappa u_{xx}, use forward Euler in time and centered differences in space: \frac{U_i^{n+1}-U_i^n}{\Delta t} = \kappa\frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. Thus U_i^{n+1}=U_i^n+\mu(U_{i-1}^n-2U_i^n+U_{i+1}^n), where \mu=\frac{\kappa\Delta t}{\Delta x^2}. Von Neumann analysis with U_i^n=\xi^ne^{i\theta i} gives \xi=1-4\mu\sin^2(\theta/2). Stability requires |\xi|\le1, so in one dimension \mu\le\frac12.
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 12.53
Problem formulation.
Prove a discrete maximum principle for the explicit heat scheme under \mu\le1/2.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For u_t=\kappa u_{xx}, use forward Euler in time and centered differences in space: \frac{U_i^{n+1}-U_i^n}{\Delta t} = \kappa\frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. Thus U_i^{n+1}=U_i^n+\mu(U_{i-1}^n-2U_i^n+U_{i+1}^n), where \mu=\frac{\kappa\Delta t}{\Delta x^2}. Von Neumann analysis with U_i^n=\xi^ne^{i\theta i} gives \xi=1-4\mu\sin^2(\theta/2). Stability requires |\xi|\le1, so in one dimension \mu\le\frac12.
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 12.54
Problem formulation.
Analyze numerical dispersion for the centered wave-equation scheme.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For u_{tt}=c^2u_{xx}, use centered differences: \frac{U_i^{n+1}-2U_i^n+U_i^{n-1}}{\Delta t^2} = c^2\frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. Thus U_i^{n+1} = 2U_i^n-U_i^{n-1} + \lambda^2(U_{i-1}^n-2U_i^n+U_{i+1}^n), where \lambda=\frac{c\Delta t}{\Delta x}. Von Neumann analysis gives the CFL condition \lambda\le1.
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 12.55
Problem formulation.
Derive the artificial viscosity term in the modified equation of Lax–Friedrichs.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.56
Problem formulation.
Explain total variation diminishing schemes and derive a flux-limiter form.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.57
Problem formulation.
Derive the basic idea of WENO reconstruction for a scalar conservation law.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.58
Problem formulation.
Explain entropy solutions and entropy-stable numerical fluxes.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.59
Problem formulation.
Prove an H^1-error estimate for linear finite elements applied to Poisson’s equation.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For Poisson’s equation -\Delta u=f on a square grid, the five-point discretization is -\Delta_hU_{i,j} = \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2}. The discrete equation is \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2} = f_{i,j}. The truncation error is O(h^2) for smooth u.
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 12.60
Problem formulation.
Use the Aubin–Nitsche duality argument to obtain an L^2-error estimate.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.61
Problem formulation.
Analyze the effect of underintegration on finite-element consistency.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.62
Problem formulation.
State local convergence conditions for Newton’s method applied to nonlinear PDE discretizations.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.63
Problem formulation.
Design a continuation method for a nonlinear elliptic PDE with parameter \lambda.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.64
Problem formulation.
Derive a stable time-stepping method for u_t=\epsilon^2\Delta u+u-u^3.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.65
Problem formulation.
Derive a Strang splitting method for the nonlinear Schrodinger equation.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.66
Problem formulation.
Formulate a Fourier pseudospectral method for viscous Burgers equation and include dealiasing.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.67
Problem formulation.
Investigate conditioning of Chebyshev discretizations for elliptic PDEs.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.68
Problem formulation.
Design a spectral filter for stabilizing a pseudospectral nonlinear PDE solver.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.69
Problem formulation.
Derive the Cox–de Boor recursion for B-splines.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.70
Problem formulation.
Derive the stiffness matrix entries for an isogeometric Poisson discretization.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For Poisson’s equation -\Delta u=f on a square grid, the five-point discretization is -\Delta_hU_{i,j} = \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2}. The discrete equation is \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2} = f_{i,j}. The truncation error is O(h^2) for smooth u.
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 12.71
Problem formulation.
Explain how knot multiplicity affects B-spline smoothness.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.72
Problem formulation.
Analyze stability of an upwind DG method for linear advection.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.73
Problem formulation.
Derive a symmetric interior penalty DG method for Poisson’s equation.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For Poisson’s equation -\Delta u=f on a square grid, the five-point discretization is -\Delta_hU_{i,j} = \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2}. The discrete equation is \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2} = f_{i,j}. The truncation error is O(h^2) for smooth u.
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 12.74
Problem formulation.
Design a multigrid V-cycle for the finite-difference Poisson equation.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For Poisson’s equation -\Delta u=f on a square grid, the five-point discretization is -\Delta_hU_{i,j} = \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2}. The discrete equation is \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2} = f_{i,j}. The truncation error is O(h^2) for smooth u.
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 12.75
Problem formulation.
Explain additive Schwarz preconditioning for PDE systems.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.76
Problem formulation.
Study full approximation scheme multigrid for nonlinear elliptic PDEs.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.77
Problem formulation.
Design physics-based preconditioners for Newton–Krylov discretizations of nonlinear PDEs.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.78
Problem formulation.
Develop residual estimators for adaptive finite elements applied to nonlinear elliptic PDEs.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.79
Problem formulation.
Derive a dual-weighted residual estimator for a quantity of interest.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.80
Problem formulation.
Study entropy-stable finite-volume or DG schemes for systems of conservation laws.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.81
Problem formulation.
Develop a fifth-order WENO method and analyze its behavior near smooth extrema.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.82
Problem formulation.
Design a limiter or artificial-viscosity strategy for DG shock capturing.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.83
Problem formulation.
Discuss Fourier pseudospectral methods for incompressible turbulence and the role of dealiasing.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.84
Problem formulation.
Derive a projection method for the incompressible Navier–Stokes equations.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.85
Problem formulation.
Design energy-stable schemes for Cahn–Hilliard or Allen–Cahn equations.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.86
Problem formulation.
Study numerical methods preserving mass, energy, positivity, or symplectic structure for PDEs.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.87
Problem formulation.
Formulate a space-time Galerkin method for a parabolic PDE.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.88
Problem formulation.
Develop a spectral element method combining high-order polynomial elements and Gauss–Lobatto quadrature.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.89
Problem formulation.
Design an hp-adaptive finite-element method for elliptic PDEs with singularities.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.90
Problem formulation.
Study isogeometric discretization and Newton linearization for nonlinear PDEs.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.91
Problem formulation.
Compare finite difference, spectral, and Galerkin methods for fractional Laplacian PDEs.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
For Poisson’s equation -\Delta u=f on a square grid, the five-point discretization is -\Delta_hU_{i,j} = \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2}. The discrete equation is \frac{ 4U_{i,j}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1} }{h^2} = f_{i,j}. The truncation error is O(h^2) for smooth u.
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 12.92
Problem formulation.
Develop a POD–Galerkin reduced-order model for a nonlinear parabolic PDE.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.93
Problem formulation.
Compare Galerkin finite elements with residual-minimizing neural solvers for nonlinear PDEs.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.94
Problem formulation.
Study stochastic Galerkin or stochastic collocation methods for PDEs with random coefficients.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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 12.95
Problem formulation.
Design reproducible benchmarks comparing finite differences, finite volumes, finite elements, splines, Fourier spectral, Chebyshev spectral, and DG methods.
Method.
Use finite differences, method of lines, CFL analysis, weak formulations, flux balances, and Fourier/von Neumann stability.
Detailed solution and justification.
The PDE is converted to algebraic equations by replacing derivatives with differences, finite-volume fluxes, finite-element weak forms, or spectral derivatives. Stability is typically checked by a Fourier mode U_i^n=\xi^n e^{i\theta i}. The method is stable when |\xi(\theta)|\le1 for every resolvable wavenumber \theta. Consistency follows from Taylor expansion and convergence follows from stability plus consistency.
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.
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