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

The first two modified stages giveThe correction stage isUsing $A_x{A}_y=A_y{A}_x$,Therefore

On the common eigenvector $v^{(p,q)}$, the eigenvalue of $D$ isSet $a=-\mu {\lambda }_p>0$ and $b=-\mu {\lambda }_q>0$. ThenBoth numerator and denominator are positive, andHencefor every mode and every $\mu >0$. Orthogonal diagonalization now gives $\Vert D{\Vert }_2<1$, so the method is stable without a time-step restriction. This is the unconditionally stable corrected directional diffusion splitting.