Let there be $J$ interior points. The <Dirichlet discrete Laplacian> has positive values
$$
\eta_j=4\sin^2\left(\frac{j\pi}{2(J+1)}\right)
$$
for 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> is
$$
\boxed{\mu\geq0
\quad\text{or}\quad
\mu\leq-\frac1{2\sin^2(\pi/(2(J+1)))}}.
$$
If, as usual, a Courant number is restricted to nonnegative values, this again reduces to all $\mu\geq0$. The additional negative branch is a finite-grid artefact and disappears to $-\infty$ as the mesh is refined.

Solution

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

{c}

On $J$ interior Dirichlet grid points, backward Euler diffusion is stable for $\mu\geq0$ and also for the nonphysical branch $\mu\leq-[2\sin^2(\pi/(2(J+1)))]^{-1}$. The latter makes every amplification denominator at most $-1$.


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