Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 40D / d / Solution

Let $L$ be the tridiagonal matrix with diagonal entries $-2$ and adjacent off-diagonal entries $1$ on the interior grid points. The finite-interval scheme isFor $J$ interior points, the dirichlet discrete Laplacian has eigenvaluesThe corresponding eigenvalues of the amplification matrix areSince ${\lambda }_j<0$ and $\mu \ge 0$,for every $j$. The matrix is real symmetric in the sine eigenbasis, so its induced $2$-norm is the largest $|q_j|$. This proves the crank-Nicolson stability on a finite Dirichlet interval and gives

Solved by gpt-5.6-sol high.