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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-63/3/1/solution

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Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 63 / 3 / 1 / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

Put

h = Δ x

. The spatial operator is the sum of a centred

x

difference and a dissipative second-order forward advection stencil in

y

. For a smooth function, their Taylor expansions give

\frac{u ( x + h , y ) - u ( x - h , y )}{2 h} = u_{x} + \frac{h ^{2}}{6} u_{xxx} + O (h^{4}),

(1)

\frac{- 3 u ( x , y ) + 4 u ( x , y + h ) - u ( x , y + 2 h )}{2 h} = u_{y} - \frac{h ^{2}}{3} u_{yyy} - \frac{h ^{3}}{4} u_{yyyy} + O (h^{4}) .

(2)

Thus the leading spatial local truncation error is

h^{2} (u_{xxx} /6 - u_{yyy} /3)

, generally nonzero. The semidiscretization is second order in space. Time has not yet been discretized, so there is no separate time order or time-step condition at this stage.

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