Math 6610 • Analysis of Numerical Methods I

{Unconditionally stable scheme}

For the one-dimensional heat equation,
\[
  u_t=u_{xx},
\]
consider the Crank-Nicolson, central difference scheme
\[
  \frac{u_j^{n+1}-u_j^n}{\Delta t}
  =
  \frac{1}{2}
  \left(
    \frac{u_{j+1}^{n+1}-2u_j^{n+1}+u_{j-1}^{n+1}}{h^2}
    +
    \frac{u_{j+1}^{n}-2u_j^{n}+u_{j-1}^{n}}{h^2}
  \right)
\]
where \(\Delta t\) and \(h\) are temporal and spatial stepsizes, respectively, and
where \(u_j^n\) is the numerical approximation to \(u(x=jh,t=n\Delta t)\), with
\(x_0=0\) and the initial time \(t=0\). Use von Neumann stability analysis to show
that this scheme is unconditionally stable.