Math 6610 • Analysis of Numerical Methods I

{Convection-Diffusion Equation} For the convection-diffusion equation
\[
  v_t+av_x=bv_{xx},
\]
where \(a>0\) and \(b>0\), a natural Crank-Nicolson scheme is
\[
\begin{aligned}
  \frac{u_j^{n+1}-u_j^n}{\Delta t}
  &+\frac{a}{2}\left(
    \frac{u_{j+1}^n-u_{j-1}^n}{2\Delta x}
    +\frac{u_{j+1}^{n+1}-u_{j-1}^{n+1}}{2\Delta x}
  \right) \\
  &=
  \frac{b}{2}\left(
    \frac{u_{j+1}^n-2u_j^n+u_{j-1}^n}{(\Delta x)^2}
    +\frac{u_{j+1}^{n+1}-2u_j^{n+1}+u_{j-1}^{n+1}}{(\Delta x)^2}
  \right).
\end{aligned}
\]
Let \(\nu=b\Delta t/(\Delta x)^2\) and \(\mu=a\Delta t/(2\Delta x)\). Show that the
scheme is stable in the maximum norm if \(\mu\le \nu\le 1\). Also use Fourier analysis
to derive the stability condition in the \(\ell_2\) norm and show that the scheme is
always stable in the \(\ell_2\) norm. Suppose \(u\) stands for some volume fraction of a
certain species, which norm is more appropriate? Explain why this scheme is so
restrictive that it is not a good choice for many applications.