 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2011/ii/paper-3/39a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Put

τ = Δ t

and

h = Δ x

, with

h^{2} = τ / μ

for fixed

μ > 0

. Substitute a smooth exact solution of the heat equation into the Adams-Bashforth method. Define its one-step defect

R

as the left side minus the right side of the unscaled update. The centered finite difference satisfies

D_{h} u = \frac{u ( x - h , t ) - 2 u ( x , t ) + u ( x + h , t )}{h ^{2}} = u_{xx} + \frac{h ^{2}}{12} u_{xxxx} + O (h^{4}) .

u (t + τ) - u (t) = τ u_{t} + \frac{τ ^{2}}{2} u_{tt} + O (τ^{3}),

τ [\frac{3}{2} D_{h} u (t) - \frac{1}{2} D_{h} u (t - τ)] = τ u_{xx} + \frac{τ ^{2}}{2} u_{xx t} + \frac{τ h ^{2}}{12} u_{xxxx} + O (τ^{3} + τ^{2} h^{2} + τ h^{4}) .

Since

u_{t} = u_{xx}

and

u_{tt} = u_{xx t}

, the first two terms cancel, leaving

R = - \frac{τ ^{2}}{12 μ} u_{xxxx} + O (τ^{3}) = O (τ^{2}) .

This is the requested local error of the unscaled update. If “local truncation error” denotes the defect divided by

Δ t

, it is instead

O (Δ t)

under fixed

μ

: the centered spatial error is then

O (h^{2}) = O (Δ t)

. The scheme is second order in time at fixed spatial discretization, which is a separate order statement.

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