Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 41C / a

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Solution

At an interior grid point, the five-point finite difference method gives

\nabla^{2} u (ih, jh, t) \approx \frac{1}{h ^{2}} (u_{i + 1, j} + u_{i - 1, j} + u_{i, j + 1} + u_{i, j - 1} - 4 u_{ij}) .

Let

T = - 2 1 1 - 2 ⋱ ⋱ ⋱ 1 1 - 2 \in R^{m \times m},

where the zero Dirichlet boundary values are already incorporated. Stack the unknowns with the

y

index varying fastest. Using the Kronecker product, define

A_{x} = T \otimes I_{m}, A_{y} = I_{m} \otimes T .

Then the semidiscrete equation is

\frac{d u}{d t} = \frac{1}{h ^{2}} (A_{x} + A_{y}) u, u (t) \in R^{m^{2}} .

Moreover,

A_{x} A_{y} = (T \otimes I) (I \otimes T) = T \otimes T = (I \otimes T) (T \otimes I) = A_{y} A_{x} .

Thus the directional matrices commute, as in the five-point Dirichlet Laplacian as a Kronecker sum.

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