Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 341 / 3 / c / Solution

Let there be $J$ interior points. The Dirichlet discrete Laplacian has positive valuesfor the eigenvalues of its negative, and the amplification eigenvalues are $(1+\mu {\eta }_j{)}^{-1}$. Therefore the Backward Euler diffusion stability on a finite Dirichlet interval isIf, as usual, a Courant number is restricted to nonnegative values, this again reduces to all $\mu \ge 0$. The additional negative branch is a finite-grid artefact and disappears to $-\infty$ as the mesh is refined.