Math 6610 • Analysis of Numerical Methods I

{Heat Equation Stability:}

\partitem{a} Consider the initial value problem for the constant-coefficient
diffusion equation
\[
  v_t=\beta v_{xx},\qquad t>0
\]
with initial data \(v(x,0)=f(x)\). The Crank-Nicolson scheme for this problem is:
\[
  \frac{u_j^{n+1}-u_j^n}{k}
  =
  \frac{\beta}{2h^2}
  \left\{
    u_{j-1}^{n+1}-2u_j^{n+1}+u_{j+1}^{n+1}
    +u_{j-1}^{n}-2u_j^{n}+u_{j+1}^{n}
  \right\}.
\]
Analyze the 2-norm stability of this scheme and show that the scheme is stable
for any choice of \(k>0\) and \(h>0\).

\partitem{b} Consider the variable coefficient diffusion equation
\[
  v_t=(\beta v_x)_x,\qquad 0<x<1,\quad t>0
\]
with Dirichlet boundary conditions
\[
  v(0,t)=0,\qquad v(1,t)=0
\]
and initial data \(v(x,0)=f(x)\). Assume that \(\beta(x)\geq\beta_0>0\), and that
\(\beta(x)\) is smooth. Let \(\beta_{j+1/2}=\beta(x_{j+1/2})\). The
Crank-Nicolson scheme for this problem is:
\[
  \begin{aligned}
  \frac{u_j^{n+1}-u_j^n}{k}
  &=
  \frac{1}{2h^2}
  \bigl\{
    \beta_{j-1/2}u_{j-1}^{n+1}
    -(\beta_{j-1/2}+\beta_{j+1/2})u_j^{n+1}
    +\beta_{j+1/2}u_{j+1}^{n+1} \\
  &\qquad
    +\beta_{j-1/2}u_{j-1}^{n}
    -(\beta_{j-1/2}+\beta_{j+1/2})u_j^{n}
    +\beta_{j+1/2}u_{j+1}^{n}
  \bigr\}.
  \end{aligned}
\]
Analyze the 2-norm stability of this scheme for solving this initial boundary
value problem. Do not neglect the fact that there are boundary conditions!

\newpage

\textbf{8) Numerical Methods for ODEs:} Consider the Linear Multistep Method
\[
  y_{n+2}-\frac{4}{3}y_{n+1}+\frac{1}{3}y_n=\frac{2}{3}h f_{n+2}
\]
for solving an initial value problem \(y'=f(y,x)\), \(y(0)=\eta\). You may assume
that \(f\) is Lipschitz continuous with respect to \(y\) uniformly for all \(x\).

\partitem{a} Analyze the consistency, stability, accuracy, and convergence
properties of this method.

\partitem{b} Would it be more reasonable to use this method or Euler's method
for the initial value problem:
\[
  y'=-10^8[y-\cos(x)]-\sin(x),\qquad y(0)=1?
\]
Justify your answer.