Math 6610 • Analysis of Numerical Methods I

{Heat Equation Stability}

Consider the variable coefficient diffusion equation
\[
  v_t=(\beta(x)v_x)_x,\qquad 0<x<1,\quad t>0
\]
with Dirichlet boundary conditions
\[
  v(0,t)=0,\qquad v(1,t)=0
\]
and initial data \(v(x,0)=f(x)\). Assume that \(\beta(x)\ge\beta_0>0\), 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 THE PROBLEM HAS VARIABLE COEFFICIENTS AND THAT
THERE ARE BOUNDARY CONDITIONS AT 0 AND 1!

\newpage