Math 6610 • Analysis of Numerical Methods I

{Convection-Diffusion Equation}
For the convection-diffusion equation
\[
  u_t+au_x=bu_{xx},
\]
where \(a>0\) and \(b>0\), a natural Crank-Nicolson scheme is
\[
\begin{aligned}
  \frac{U_j^{n+1}-U_j^n}{\Delta t}
  &+\frac{a}{2}\left(
    \frac{U_{j+1}^n-U_{j-1}^n}{2\Delta x}
    +
    \frac{U_{j+1}^{n+1}-U_{j-1}^{n+1}}{2\Delta x}
  \right)\\
  &=
  \frac{b}{2}\left(
    \frac{U_{j+1}^n-2U_j^n+U_{j-1}^n}{(\Delta x)^2}
    +
    \frac{U_{j+1}^{n+1}-2U_j^{n+1}+U_{j-1}^{n+1}}{(\Delta x)^2}
  \right).
\end{aligned}
\]
Denote \(\nu=b\Delta t/(\Delta x)^2\) and \(\mu=a\Delta t/(2\Delta x)\); show that the
scheme is stable in the max 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 chemical
concentration, which norm is more appropriate? Explain why this scheme is so
restrictive that it is not often used.