 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-341/3/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

The original PDF puts both neighbouring values at time

n + 1

. Thus the actual scheme is Backward Euler method in time with a centered three-point approximation to the second spatial derivative. The old-time neighbours in the TeX transcription describe a different method.

Let

h = Δ x

and

k = Δ t

. Insert a sufficiently smooth exact solution and divide by

k

. Expanding at

(x_{m}, t_{n + 1})

gives

\frac{u ( x _{m} , t _{n + 1} ) - u ( x _{m} , t _{n} )}{k} - \frac{u ( x _{m} - h , t _{n + 1} ) - 2 u ( x _{m} , t _{n + 1} ) + u ( x _{m} + h , t _{n + 1} )}{h ^{2}} = - \frac{k}{2} u_{tt} - \frac{h ^{2}}{12} u_{xxxx} + O (k^{2} + h^{4}) .

The PDE cancels the

u_{t} - u_{xx}

term. Therefore the normalized local truncation error is

O (k + h^{2})

and

first order in time and second order in space .

For compatible smooth data with exact nodal initialization, the stability of a numerical method established below gives global error

O (k + h^{2})

on fixed time intervals, by summing the one-step defects with the contraction bound. Thus no relation between

k

and

h

is needed for stability; choosing

k = O (h^{2})

makes both consistency contributions

O (h^{2})

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