Initial-Value Problems for Ordinary Differential Equations

Problem formulation.

Derive the integral form y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Derive explicit Euler by approximating the integral with the left endpoint rule.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

The exact integral form is y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. Explicit Euler approximates the integral by the left endpoint rule: \int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s = h f(t_n,y(t_n))+O(h^2). Thus y_{n+1}=y_n+h f(t_n,y_n). Taylor expansion gives the local truncation error \tau_{n+1} = \frac{h^2}{2}y''(\xi_n), so the method has local order 2 and global order 1.

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.

Problem formulation.

Derive backward Euler by approximating the integral with the right endpoint rule.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

The exact integral form is y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. Explicit Euler approximates the integral by the left endpoint rule: \int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s = h f(t_n,y(t_n))+O(h^2). Thus y_{n+1}=y_n+h f(t_n,y_n). Taylor expansion gives the local truncation error \tau_{n+1} = \frac{h^2}{2}y''(\xi_n), so the method has local order 2 and global order 1.

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.

Problem formulation.

Derive the trapezoidal method for IVPs.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Prove the local truncation error formula for explicit Euler.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

The exact integral form is y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. Explicit Euler approximates the integral by the left endpoint rule: \int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s = h f(t_n,y(t_n))+O(h^2). Thus y_{n+1}=y_n+h f(t_n,y_n). Taylor expansion gives the local truncation error \tau_{n+1} = \frac{h^2}{2}y''(\xi_n), so the method has local order 2 and global order 1.

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.

Problem formulation.

Apply one explicit Euler step to y'=t+y,\qquad y(0)=1, with h=0.1.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

The exact integral form is y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. Explicit Euler approximates the integral by the left endpoint rule: \int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s = h f(t_n,y(t_n))+O(h^2). Thus y_{n+1}=y_n+h f(t_n,y_n). Taylor expansion gives the local truncation error \tau_{n+1} = \frac{h^2}{2}y''(\xi_n), so the method has local order 2 and global order 1.

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.

Problem formulation.

Write the four stages for one RK4 step applied to y'=y,\qquad y(0)=1.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

For classical RK4, k_1=f(t_n,y_n), k_2=f\left(t_n+\frac h2,y_n+\frac h2k_1\right), \quad k_3=f\left(t_n+\frac h2,y_n+\frac h2k_2\right), k_4=f(t_n+h,y_n+hk_3), y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4). Applied to y'=\lambda y, z=h\lambda, this gives R(z)=1+z+\frac{z^2}{2}+\frac{z^3}{6}+\frac{z^4}{24}. The method matches the Taylor series through order h^4, hence local error O(h^5) and global error O(h^4).

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.

Problem formulation.

Write the explicit formulas for midpoint RK2.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Write the explicit formulas for Heun’s RK2 method.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Write the explicit formulas for the classical third-order RK method.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Write the explicit formulas for classical RK4.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

For classical RK4, k_1=f(t_n,y_n), k_2=f\left(t_n+\frac h2,y_n+\frac h2k_1\right), \quad k_3=f\left(t_n+\frac h2,y_n+\frac h2k_2\right), k_4=f(t_n+h,y_n+hk_3), y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4). Applied to y'=\lambda y, z=h\lambda, this gives R(z)=1+z+\frac{z^2}{2}+\frac{z^3}{6}+\frac{z^4}{24}. The method matches the Taylor series through order h^4, hence local error O(h^5) and global error O(h^4).

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.

Problem formulation.

Write the explicit formulas for the Kutta 3/8 fourth-order method.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

For y'=\lambda y, derive the explicit Euler stability function.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Show that explicit Euler is stable when |1+h\lambda|\le1.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Derive the backward Euler stability function.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Explain stiffness in one paragraph.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Explain how an embedded RK pair estimates local error.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Write the AB2 formula.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Linear multistep methods approximate the integral or derivative using values from several time levels. For example, AB2 is obtained by integrating the linear interpolant of f: y_{n+1}=y_n+\frac h2(3f_n-f_{n-1}). BDF2 differentiates the quadratic interpolant through y_{n-1},y_n,y_{n+1}: \frac{3y_{n+1}-4y_n+y_{n-1}}{2h} = f(t_{n+1},y_{n+1}). Its characteristic polynomial satisfies the root condition: 3\xi^2-4\xi+1=(\xi-1)(3\xi-1), so it is zero-stable.

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.

Problem formulation.

Write the BDF2 formula.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Linear multistep methods approximate the integral or derivative using values from several time levels. For example, AB2 is obtained by integrating the linear interpolant of f: y_{n+1}=y_n+\frac h2(3f_n-f_{n-1}). BDF2 differentiates the quadratic interpolant through y_{n-1},y_n,y_{n+1}: \frac{3y_{n+1}-4y_n+y_{n-1}}{2h} = f(t_{n+1},y_{n+1}). Its characteristic polynomial satisfies the root condition: 3\xi^2-4\xi+1=(\xi-1)(3\xi-1), so it is zero-stable.

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.

Problem formulation.

Explain the basic idea of Chebyshev collocation for an IVP.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Prove the referenced result.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Prove the referenced result.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

The exact integral form is y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. Explicit Euler approximates the integral by the left endpoint rule: \int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s = h f(t_n,y(t_n))+O(h^2). Thus y_{n+1}=y_n+h f(t_n,y_n). Taylor expansion gives the local truncation error \tau_{n+1} = \frac{h^2}{2}y''(\xi_n), so the method has local order 2 and global order 1.

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.

Problem formulation.

Prove the referenced result.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Prove the referenced result.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Prove the referenced result.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Verify the order-four conditions for classical RK4.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

For classical RK4, k_1=f(t_n,y_n), k_2=f\left(t_n+\frac h2,y_n+\frac h2k_1\right), \quad k_3=f\left(t_n+\frac h2,y_n+\frac h2k_2\right), k_4=f(t_n+h,y_n+hk_3), y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4). Applied to y'=\lambda y, z=h\lambda, this gives R(z)=1+z+\frac{z^2}{2}+\frac{z^3}{6}+\frac{z^4}{24}. The method matches the Taylor series through order h^4, hence local error O(h^5) and global error O(h^4).

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.

Problem formulation.

Verify that the Kutta 3/8 method satisfies fourth-order conditions.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Derive the embedded error estimate for the RKF45 formulas given in the chapter.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Derive the embedded error estimate for the Dormand–Prince 5(4) formulas.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Derive the adaptive step-size formula h_{\mathrm{new}}=h\,sE^{-1/(p+1)}.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Prove the referenced result.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Derive the RK4 stability polynomial R(z)=1+z+\frac{z^2}{2}+\frac{z^3}{6}+\frac{z^4}{24}.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

For classical RK4, k_1=f(t_n,y_n), k_2=f\left(t_n+\frac h2,y_n+\frac h2k_1\right), \quad k_3=f\left(t_n+\frac h2,y_n+\frac h2k_2\right), k_4=f(t_n+h,y_n+hk_3), y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4). Applied to y'=\lambda y, z=h\lambda, this gives R(z)=1+z+\frac{z^2}{2}+\frac{z^3}{6}+\frac{z^4}{24}. The method matches the Taylor series through order h^4, hence local error O(h^5) and global error O(h^4).

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.

Problem formulation.

Prove the referenced result.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Show that the trapezoidal rule has stability function R(z)=\frac{1+z/2}{1-z/2}.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Derive AB2 by integrating a linear interpolant of f.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Linear multistep methods approximate the integral or derivative using values from several time levels. For example, AB2 is obtained by integrating the linear interpolant of f: y_{n+1}=y_n+\frac h2(3f_n-f_{n-1}). BDF2 differentiates the quadratic interpolant through y_{n-1},y_n,y_{n+1}: \frac{3y_{n+1}-4y_n+y_{n-1}}{2h} = f(t_{n+1},y_{n+1}). Its characteristic polynomial satisfies the root condition: 3\xi^2-4\xi+1=(\xi-1)(3\xi-1), so it is zero-stable.

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.

Problem formulation.

Derive the two-step Adams–Moulton formula.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Linear multistep methods approximate the integral or derivative using values from several time levels. For example, AB2 is obtained by integrating the linear interpolant of f: y_{n+1}=y_n+\frac h2(3f_n-f_{n-1}). BDF2 differentiates the quadratic interpolant through y_{n-1},y_n,y_{n+1}: \frac{3y_{n+1}-4y_n+y_{n-1}}{2h} = f(t_{n+1},y_{n+1}). Its characteristic polynomial satisfies the root condition: 3\xi^2-4\xi+1=(\xi-1)(3\xi-1), so it is zero-stable.

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.

Problem formulation.

Derive BDF2 by differentiating the quadratic interpolant through y_{n+1}, y_n, and y_{n-1}.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Linear multistep methods approximate the integral or derivative using values from several time levels. For example, AB2 is obtained by integrating the linear interpolant of f: y_{n+1}=y_n+\frac h2(3f_n-f_{n-1}). BDF2 differentiates the quadratic interpolant through y_{n-1},y_n,y_{n+1}: \frac{3y_{n+1}-4y_n+y_{n-1}}{2h} = f(t_{n+1},y_{n+1}). Its characteristic polynomial satisfies the root condition: 3\xi^2-4\xi+1=(\xi-1)(3\xi-1), so it is zero-stable.

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.

Problem formulation.

Check the root condition for BDF2.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Linear multistep methods approximate the integral or derivative using values from several time levels. For example, AB2 is obtained by integrating the linear interpolant of f: y_{n+1}=y_n+\frac h2(3f_n-f_{n-1}). BDF2 differentiates the quadratic interpolant through y_{n-1},y_n,y_{n+1}: \frac{3y_{n+1}-4y_n+y_{n-1}}{2h} = f(t_{n+1},y_{n+1}). Its characteristic polynomial satisfies the root condition: 3\xi^2-4\xi+1=(\xi-1)(3\xi-1), so it is zero-stable.

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.

Problem formulation.

Show that y''=f_t+f_yf.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Derive the collocation equations for a polynomial u(t) on one step.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Derive the time scaling \frac{\dd}{\dd t}=\frac{2}{T-t_0}\frac{\dd}{\dd x}.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Write the Chebyshev collocation equations for the scalar IVP y'=y,\qquad y(0)=1 on [0,1].

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Explain how RK stages can be used to construct dense output.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Describe how to locate an event g(t,y(t))=0 inside a step.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Design a work-precision experiment comparing Euler, RK2, RK4, and RK45.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

The exact integral form is y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. Explicit Euler approximates the integral by the left endpoint rule: \int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s = h f(t_n,y(t_n))+O(h^2). Thus y_{n+1}=y_n+h f(t_n,y_n). Taylor expansion gives the local truncation error \tau_{n+1} = \frac{h^2}{2}y''(\xi_n), so the method has local order 2 and global order 1.

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.

Problem formulation.

Introduce rooted trees and derive Runge–Kutta order conditions up to order four.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

For classical RK4, k_1=f(t_n,y_n), k_2=f\left(t_n+\frac h2,y_n+\frac h2k_1\right), \quad k_3=f\left(t_n+\frac h2,y_n+\frac h2k_2\right), k_4=f(t_n+h,y_n+hk_3), y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4). Applied to y'=\lambda y, z=h\lambda, this gives R(z)=1+z+\frac{z^2}{2}+\frac{z^3}{6}+\frac{z^4}{24}. The method matches the Taylor series through order h^4, hence local error O(h^5) and global error O(h^4).

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.

Problem formulation.

Study the order conditions required for an explicit fifth-order Runge–Kutta method.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

For classical RK4, k_1=f(t_n,y_n), k_2=f\left(t_n+\frac h2,y_n+\frac h2k_1\right), \quad k_3=f\left(t_n+\frac h2,y_n+\frac h2k_2\right), k_4=f(t_n+h,y_n+hk_3), y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4). Applied to y'=\lambda y, z=h\lambda, this gives R(z)=1+z+\frac{z^2}{2}+\frac{z^3}{6}+\frac{z^4}{24}. The method matches the Taylor series through order h^4, hence local error O(h^5) and global error O(h^4).

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.

Problem formulation.

Explain the first-same-as-last property of Dormand–Prince methods and its computational advantage.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Analyze how embedded RK pairs share stages to produce two orders at once.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Derive a proportional-integral step-size controller for adaptive RK methods.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Numerically plot the absolute stability region of RK4 and determine its intersection with the negative real axis.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

For classical RK4, k_1=f(t_n,y_n), k_2=f\left(t_n+\frac h2,y_n+\frac h2k_1\right), \quad k_3=f\left(t_n+\frac h2,y_n+\frac h2k_2\right), k_4=f(t_n+h,y_n+hk_3), y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4). Applied to y'=\lambda y, z=h\lambda, this gives R(z)=1+z+\frac{z^2}{2}+\frac{z^3}{6}+\frac{z^4}{24}. The method matches the Taylor series through order h^4, hence local error O(h^5) and global error O(h^4).

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.

Problem formulation.

State the first and second Dahlquist barriers and explain their implications.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Prove that the trapezoidal rule is A-stable but not L-stable.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Define L-stability and show that backward Euler is L-stable.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Analyze explicit Euler, backward Euler, and trapezoidal rule on y'=-1000(y-\cos t)-\sin t.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

The exact integral form is y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. Explicit Euler approximates the integral by the left endpoint rule: \int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s = h f(t_n,y(t_n))+O(h^2). Thus y_{n+1}=y_n+h f(t_n,y_n). Taylor expansion gives the local truncation error \tau_{n+1} = \frac{h^2}{2}y''(\xi_n), so the method has local order 2 and global order 1.

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.

Problem formulation.

Derive Newton’s method for solving one backward Euler step.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

The exact integral form is y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. Explicit Euler approximates the integral by the left endpoint rule: \int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s = h f(t_n,y(t_n))+O(h^2). Thus y_{n+1}=y_n+h f(t_n,y_n). Taylor expansion gives the local truncation error \tau_{n+1} = \frac{h^2}{2}y''(\xi_n), so the method has local order 2 and global order 1.

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.

Problem formulation.

Discuss Jacobian reuse and simplified Newton methods for stiff IVP solvers.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Derive a variable-step BDF2 formula.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Linear multistep methods approximate the integral or derivative using values from several time levels. For example, AB2 is obtained by integrating the linear interpolant of f: y_{n+1}=y_n+\frac h2(3f_n-f_{n-1}). BDF2 differentiates the quadratic interpolant through y_{n-1},y_n,y_{n+1}: \frac{3y_{n+1}-4y_n+y_{n-1}}{2h} = f(t_{n+1},y_{n+1}). Its characteristic polynomial satisfies the root condition: 3\xi^2-4\xi+1=(\xi-1)(3\xi-1), so it is zero-stable.

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.

Problem formulation.

Construct an AB2–AM2 predictor–corrector method and analyze its order.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Explain the Nordsieck representation for variable-order multistep methods.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Prove the referenced result.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Study Radau IIA methods and explain why they are useful for stiff problems.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Study Lobatto collocation methods and their relation to symplectic integration.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Derive symplectic Euler for a Hamiltonian system and compare it with explicit Euler.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

The exact integral form is y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. Explicit Euler approximates the integral by the left endpoint rule: \int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s = h f(t_n,y(t_n))+O(h^2). Thus y_{n+1}=y_n+h f(t_n,y_n). Taylor expansion gives the local truncation error \tau_{n+1} = \frac{h^2}{2}y''(\xi_n), so the method has local order 2 and global order 1.

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.

Problem formulation.

Derive the Stormer–Verlet method for second-order mechanical systems.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Construct the Chebyshev differentiation matrix and use it to solve a linear scalar IVP by collocation.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Design a numerical experiment comparing RK4 and Chebyshev collocation for an analytic solution on [0,1].

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

For classical RK4, k_1=f(t_n,y_n), k_2=f\left(t_n+\frac h2,y_n+\frac h2k_1\right), \quad k_3=f\left(t_n+\frac h2,y_n+\frac h2k_2\right), k_4=f(t_n+h,y_n+hk_3), y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4). Applied to y'=\lambda y, z=h\lambda, this gives R(z)=1+z+\frac{z^2}{2}+\frac{z^3}{6}+\frac{z^4}{24}. The method matches the Taylor series through order h^4, hence local error O(h^5) and global error O(h^4).

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.

Problem formulation.

Derive one correction sweep of spectral deferred correction for an explicit Euler base method.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

The exact integral form is y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. Explicit Euler approximates the integral by the left endpoint rule: \int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s = h f(t_n,y(t_n))+O(h^2). Thus y_{n+1}=y_n+h f(t_n,y_n). Taylor expansion gives the local truncation error \tau_{n+1} = \frac{h^2}{2}y''(\xi_n), so the method has local order 2 and global order 1.

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.

Problem formulation.

Formulate an IVP solver that minimizes the residual of the differential equation over a polynomial trial space.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Derive a fourth-order Hermite–Obreschkoff method using endpoint derivatives.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Develop B-series for Runge–Kutta methods and prove composition rules.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

For classical RK4, k_1=f(t_n,y_n), k_2=f\left(t_n+\frac h2,y_n+\frac h2k_1\right), \quad k_3=f\left(t_n+\frac h2,y_n+\frac h2k_2\right), k_4=f(t_n+h,y_n+hk_3), y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4). Applied to y'=\lambda y, z=h\lambda, this gives R(z)=1+z+\frac{z^2}{2}+\frac{z^3}{6}+\frac{z^4}{24}. The method matches the Taylor series through order h^4, hence local error O(h^5) and global error O(h^4).

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.

Problem formulation.

Study algebraic stability of implicit Runge–Kutta methods and its connection with contractivity.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

For classical RK4, k_1=f(t_n,y_n), k_2=f\left(t_n+\frac h2,y_n+\frac h2k_1\right), \quad k_3=f\left(t_n+\frac h2,y_n+\frac h2k_2\right), k_4=f(t_n+h,y_n+hk_3), y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4). Applied to y'=\lambda y, z=h\lambda, this gives R(z)=1+z+\frac{z^2}{2}+\frac{z^3}{6}+\frac{z^4}{24}. The method matches the Taylor series through order h^4, hence local error O(h^5) and global error O(h^4).

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.

Problem formulation.

Investigate order reduction of high-order methods on stiff problems with boundary or forcing layers.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Derive exponential Euler and exponential Runge–Kutta methods for semilinear problems.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

The exact integral form is y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. Explicit Euler approximates the integral by the left endpoint rule: \int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s = h f(t_n,y(t_n))+O(h^2). Thus y_{n+1}=y_n+h f(t_n,y_n). Taylor expansion gives the local truncation error \tau_{n+1} = \frac{h^2}{2}y''(\xi_n), so the method has local order 2 and global order 1.

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.

Problem formulation.

Develop an implicit–explicit method for y'=F(t,y)+G(t,y), where G is stiff.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Study Rosenbrock–W methods for stiff systems and derive their linear stage solves.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Analyze Krylov approximations for computing \varphi-functions in exponential integrators.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Study Parareal or PFASST methods for parallel-in-time integration.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Prove a rigorous spectral convergence theorem for Chebyshev collocation applied to a linear analytic IVP.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Develop an hp-version time finite element method for IVPs and compare it with Chebyshev collocation.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Compare adaptive RK solvers with residual-minimizing polynomial or neural trial functions for IVPs.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Study interval Taylor methods for validated integration of ODEs.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Analyze long-time energy behavior of symplectic methods for Hamiltonian systems.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Design an adaptive stiffness detector that switches between explicit and implicit methods.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.

Problem formulation.

Extend the discussion to differential-algebraic equations and explain index-related difficulties.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Discuss how IVP solvers change for delay differential equations.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Compare classical IVP solvers with predictor–corrector methods for Caputo fractional differential equations.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Analyze mixed-precision time stepping with high-precision error estimation.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Start from y(t_{n+1})=y(t_n)+\int_{t_n}^{t_{n+1}}f(s,y(s))\,\dd s. One-step methods approximate this integral by quadrature or collocation. Stability is tested with y'=\lambda y, which produces an amplification factor y_{n+1}=R(h\lambda)y_n. The method is stable for those z=h\lambda satisfying |R(z)|\le1. Accuracy follows by Taylor expansion and comparison with the exact solution series.

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.

Problem formulation.

Design reproducible benchmarks comparing RK4, RK45, BDF, Radau, Chebyshev collocation, and SDC methods.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

For classical RK4, k_1=f(t_n,y_n), k_2=f\left(t_n+\frac h2,y_n+\frac h2k_1\right), \quad k_3=f\left(t_n+\frac h2,y_n+\frac h2k_2\right), k_4=f(t_n+h,y_n+hk_3), y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4). Applied to y'=\lambda y, z=h\lambda, this gives R(z)=1+z+\frac{z^2}{2}+\frac{z^3}{6}+\frac{z^4}{24}. The method matches the Taylor series through order h^4, hence local error O(h^5) and global error O(h^4).

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.

Problem formulation.

Design a decision system that selects an IVP solver based on stiffness, smoothness, accuracy requirements, and event structure.

Method.

Use the exact integral form of the IVP, Taylor expansion, Runge–Kutta order conditions, linear stability analysis, and collocation equations.

Detailed solution and justification.

Apply the method to the test equation y'=\lambda y,\qquad z=h\lambda. The numerical update has the form y_{n+1}=R(z)y_n. Absolute stability requires |R(z)|\le1. For explicit Euler, R(z)=1+z, so the stability region is |1+z|\le1. For backward Euler, R(z)=\frac{1}{1-z}, which is A-stable because |1-z|\ge1 for \Re z\le0. Stiffness occurs when stability imposes a much smaller step size than accuracy requires.

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.