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

The original PDF puts both neighbouring values at time $n+1$. Thus the actual scheme is Backward Euler method in time with a centered three-point approximation to the second spatial derivative. The old-time neighbours in the TeX transcription describe a different method.

Let $h=\Delta x$ and $k=\Delta t$. Insert a sufficiently smooth exact solution and divide by $k$. Expanding at $(x_m,t_{n+1})$ givesThe PDE cancels the $u_t-u_{xx}$ term. Therefore the normalized local truncation error is $O(k+h^2)$ andFor compatible smooth data with exact nodal initialization, the stability of a numerical method established below gives global error $O(k+h^2)$ on fixed time intervals, by summing the one-step defects with the contraction bound. Thus no relation between $k$ and $h$ is needed for stability; choosing $k=O(h^2)$ makes both consistency contributions $O(h^2)$.

Let $T$ be the $M\times M$ tridiagonal matrix with diagonal two and adjacent entries minus one, incorporating zero Dirichlet boundary conditions. The actual PDF scheme isAn orthonormal basis of eigenvectors of $T$ has components $\sqrt{2/(M+1)}\sin (jm\pi /(M+1))$, and substitution givesThey form an orthonormal basis by the spectral theorem for real symmetric matrices. Hence $I+\mu T$ is invertible for every $\mu >0$, and the amplification eigenvalues are $g_j=(1+\mu {\lambda }_j{)}^{-1}\in (0,1)$. For the discrete L2 norm $\Vert U{\Vert }_h^2=h{\sum }_m|U_m{|}^2$,This bound has constant one independent of time step and mesh, which is the required stability of a numerical method. Therefore every positive Courant number is stable. If a forcing or local-defect sequence is added, the discrete variation-of-constants formula gives a bound by the initial norm plus the sum of those perturbation norms; this also justifies the global error conclusion in part (a).

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.