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 \(v(x,0)=f(x)\). A scheme for this problem is:
\[
  \frac{u_j^{n+1}-u_j^n}{k}
  =
  \frac{\beta}{h^2}
  \left\{u_{j-1}^{n+1}-2u_j^{n+1}+u_{j+1}^{n+1}\right\}.
\]
Analyze the 2-norm stability of this scheme. For which values of \(k>0\) and
\(h>0\) is the scheme stable? (Note that this there are no boundary conditions
here.)

\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 \(\beta_{j+1/2}=\beta(x_{j+1/2})\). A scheme for this
problem is:
\[
  \frac{u_j^{n+1}-u_j^n}{k}
  =
  \frac{1}{h^2}
  \left\{
    \beta_{j-1/2}u_{j-1}^{n+1}
    -(\beta_{j-1/2}+\beta_{j+1/2})u_j^{n+1}
    +\beta_{j+1/2}u_{j+1}^{n+1}
  \right\}.
\]
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!

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