Let $U_m^n=u(mh,nk)$ be the sampled exact solution and let $A_h$ denote the update matrix of the <Forward Euler diffusion scheme>. For $0\leq\mu\leq1/2$,
$$
(A_hv)_m=\mu v_{m-1}+(1-2\mu)v_m+\mu v_{m+1}
$$
is a convex combination, with the homogeneous boundary values included. Therefore
$$
\|A_hv\|_\infty\leq\|v\|_\infty.
$$
This is the parabolic <discrete maximum principle> and proves max-norm stability directly.

The <Taylor theorem> and the <second-order central difference> give the exact-grid residual
$$
U^{n+1}=A_hU^n+k\tau^n,
\qquad
\|\tau^n\|_\infty\leq C(k+h^2)
$$
for a sufficiently smooth solution on a fixed interval $0\leq t\leq T$. If $e^n=U^n-u^n$, then
$$
\|e^{n+1}\|_\infty\leq\|e^n\|_\infty+k\|\tau^n\|_\infty.
$$
Iteration for $nk\leq T$ yields
$$
\boxed{\|e^n\|_\infty\leq\|e^0\|_\infty+CT(k+h^2)\longrightarrow0.}
$$
\b[Thus $\mu\leq1/2$ implies convergence when the initial grid values converge to the initial data.]

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 40C / a / Solution

{c}

For the centered spatial second difference, forward Euler has the update
$$
u_m^{n+1}=\mu u_{m-1}^n+(1-2\mu)u_m^n+\mu u_{m+1}^n.
$$
It is stable for $0\leq\mu\leq1/2$. In this range the update is a convex combination, which gives a direct maximum-norm proof as well as the Fourier proof.


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