Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 341 / 3 / c

Use $N$ distinct periodic grid points $x_m=m/N$, $0\le m<N$, and $h=1/N$. Values at the two ends of the interval represent the same point, so the endpoint must not be stored twice. With indices interpreted modulo $N$, setThis replaces the Dirichlet matrix by the periodic discrete Laplacian. The normalized Fourier modes $N^{-1/2}{e}^{2\pi ijm/N}$ form an orthonormal basis and have amplification factorsEach satisfies $0<g_j\le 1$ for every $\mu >0$. The Parseval identity therefore gives $\Vert U^{n+1}{\Vert }_h\le \Vert U^n{\Vert }_h$ in the discrete L2 norm, uniformly in the grid and time step. Thus all positive Courant numbers remain stable. The constant mode has $g_0=1$, expressing preservation of the spatial mean; all nonconstant modes decay. The formula with modulo indices also handles small grids, where two neighbours can coincide.