Math 6610 • Analysis of Numerical Methods I

{Heat Equation} Consider the following heat equation problem
\[
  u_t=u_{xx},\qquad t>0,\qquad 0<x<1,
\]
\[
  u(0,t)=u(1,t)=0,\qquad t>0,
\]
\[
  u(x,0)=u_0(x),\qquad 0\le x\le1.
\]
Using the forward or backward difference for the time derivative and central
difference for the space derivative, derive the explicit and implicit schemes.
What's the stability condition for each case? Perform a Fourier analysis on both
the explicit and implicit methods and use the von Neumann condition to verify
these stability conditions.