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

Eliminating the intermediate vector gives the amplification matrixThe commuting real symmetric matrices admit simultaneous diagonalization in the orthonormal basis $v^{(p,q)}$ from the question. On this vector,Since the basis is orthonormal, stability in the discrete Euclidean norm is equivalent to $|c_{pq}|\le 1$ for every $p,q$.

PutThen $-\beta \le {\lambda }_j\le -\alpha$. The smallest denominator is $1+\mu \alpha$, while the largest numerator modulus occurs at an endpoint of this interval. The condition involving $1-\mu \alpha$ is automatic, and the other isFor $m>1$, this is equivalent toSince $\beta -\alpha =4\cos (\pi h)$, the exact finite-grid stability condition isFor $m=1$ there is only one mode and the method is stable for every $\mu >0$. A simple grid-independent condition, and the limiting condition as $h\to 0$, isThis is the stability limit of one-implicit-direction diffusion splitting.