Unconditionally stable corrected directional diffusion splitting
= Unconditionally stable corrected directional 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 $\mu>0$.