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

Solution

Let

U_{m}^{n} = u (mh, nk)

be the sampled exact solution and let

A_{h}

0 \leq μ \leq 1/2

(A_{h} v)_{m} = μ v_{m - 1} + (1 - 2 μ) v_{m} + μ v_{m + 1}

is a convex combination, with the homogeneous boundary values included. Therefore

∥ A_{h} v ∥_{\infty} \leq ∥ v ∥_{\infty} .

This is the parabolic discrete maximum principle and proves max-norm stability directly.

U^{n + 1} = A_{h} U^{n} + k τ^{n}, ∥ τ^{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 ∥ τ^{n} ∥_{\infty} .

nk \leq T

yields

∥ e^{n} ∥_{\infty} \leq ∥ e^{0} ∥_{\infty} + CT (k + h^{2}) ⟶ 0.

Thus $μ \leq 1/2$ implies convergence when the initial grid values converge to the initial data.

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