Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 40C / b / i / Solution

Sample the diffusion coefficient at edge midpoints and write

α γ = a ((i - \frac{1}{2}) h, jh), = a (ih, (j - \frac{1}{2}) h), β δ = a ((i + \frac{1}{2}) h, jh), = a (ih, (j + \frac{1}{2}) h) .

Centered differences of the two fluxes give the conservative finite-difference stencil

\frac{1}{h ^{2}} [α u_{i - 1, j} + β u_{i + 1, j} + γ u_{i, j - 1} + δ u_{i, j + 1} - (α + β + γ + δ) u_{i, j}] .

For example, the

x

part is

\frac{1}{h} [a_{i + 1/2, j} \frac{u _{i + 1, j} - u _{i, j}}{h} - a_{i - 1/2, j} \frac{u _{i, j} - u _{i - 1, j}}{h}] .

Taylor expansion about

(ih, jh)

shows that the odd powers cancel between the two face fluxes and that the result equals

\partial_{x} (a u_{x}) + O (h^{2})

. The same calculation in

y

gives total truncation error

O (h^{2}) .

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