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

This is the Backward Euler diffusion scheme with a centered second spatial difference. Taylor expansion gives first order in time and second order in space:With the parabolic scaling $\Delta t=\mu \Delta x^2$ and fixed $\mu$, the combined error is $O(\Delta x^2)$.

Insert the Fourier mode $u_m^n=G^n{e}^{im\theta }$ into the scheme. Von Neumann stability analysis givesFor every physical Courant number $\mu \ge 0$, one has $0<G(\theta )\le 1$. Thus the method is unconditionally stable: the range is $\boxed{\mu \ge 0}$.

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.