Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 41E / a

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- Solution a

Solution

Let

δ_{x}^{2} u_{m} = u_{m - 1} - 2 u_{m} + u_{m + 1}

. For an exact smooth solution of the diffusion equation, the second-order central difference gives

δ_{x}^{2} u (x, t) = h^{2} u_{xx} (x, t) + \frac{h ^{4}}{12} u_{xxxx} (x, t) + O (h^{6}) .

Since the Courant number

μ = k / h^{2}

is fixed,

h^{2} = O (k)

. Taylor expansion in time gives

u (t + k) - u (t) \frac{3}{2} μ δ_{x}^{2} u (t) - \frac{1}{2} μ δ_{x}^{2} u (t - k) = k u_{t} + \frac{k ^{2}}{2} u_{tt} + O (k^{3}), = k u_{xx} + \frac{k ^{2}}{2} u_{xx t} + O (k h^{2}) + O (k^{3}) .

Using

u_{t} = u_{xx}

and

u_{tt} = u_{xx t}

cancels the leading terms. Because

k h^{2} = O (k^{2})

, the residual, and hence the local truncation error in the convention of the question, is

O (k^{2}) .

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