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Five-point Dirichlet Laplacian as a Kronecker sum

One-implicit-direction diffusion splitting

Adding the correction
$$
u^{n+1}=\widetilde u^{n+1}+\mu A_x(u^{n+1}-u^n)
$$
gives
$$
D=(I-\mu A_x)^{-1}(I+\mu^2A_xA_y)(I-\mu A_y)^{-1}.
$$
Its common-basis eigenvalues are
$$
d_{pq}=\frac{1+\mu^2\lambda_p\lambda_q}
{(1-\mu\lambda_p)(1-\mu\lambda_q)}.
$$
Since $\lambda_p,\lambda_q<0$, one has $0<d_{pq}\leq1$ for every $\mu>0$.

Unconditionally stable corrected directional diffusion splitting

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

Unconditionally stable corrected directional diffusion splitting

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Unconditionally stable corrected directional diffusion splitting

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