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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ii/paper-2/39b/iii/solution

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Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 39B / iii / Solution by

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

The five-point stencil on the forcing satisfies

Γ_{5} f = h^{2} Δ f + \frac{h ^{4}}{12} (f_{xxxx} + f_{yyyy}) + O (h^{6}) .

(1)

Thus the correction

h^{2} Γ_{5} f /12

cancels exactly the

h^{4} Δ f /12

residual in part (i), leaving

Γ_{9} u - h^{2} f - \frac{h ^{2}}{12} Γ_{5} f = h^{6} [- \frac{u _{xxxxxx} + u _{yyyyyy}}{240} + \frac{u _{xxxx yy} + u _{xx yyyy}}{144}] + O (h^{8}) .

(2)

The modified scheme therefore has fourth-order normalized local accuracy, the same improved order as the harmonic-forcing case, now without assuming

Δ f = 0

. It need not have the same leading error constant.

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