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Unconditionally stable corrected directional diffusion splitting

Adding the correction

u^{n + 1} = u^{n + 1} + μ A_{x} (u^{n + 1} - u^{n})

gives

D = (I - μ A_{x})^{- 1} (I + μ^{2} A_{x} A_{y}) (I - μ A_{y})^{- 1} .

Its common-basis eigenvalues are

d_{pq} = \frac{1 + μ ^{2} λ _{p} λ _{q}}{( 1 - μ λ _{p} ) ( 1 - μ λ _{q} )} .

Since

λ_{p}, λ_{q} < 0

, one has

0 < d_{pq} \leq 1

for every

μ > 0

.

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Solution