Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 341 / 4

Let $L_h=D_{xx}+\alpha D_x$, where the centered second-difference matrix $D_{xx}$ is real symmetric negative definite under homogeneous Dirichlet boundary conditions, and the centered first-difference matrix $D_x$ is real skew-symmetric. HenceThe semidiscrete energy method gives $\Vert \mathbf{u}(t){\Vert }_2\le \Vert \mathbf{u}(0){\Vert }_2$ for every real $\alpha$, so the scheme is stable.

For the Cauchy problem, insert the Fourier mode $u_m=e^{im\vartheta }$. The spatial symbol isIts real part is nonpositive for every $\vartheta$, so each mode has modulus $e^{t\mathrm{Re}{\lambda }_h}\le 1$. By the discrete Fourier transform and Parseval identity, the scheme is stable in the discrete $L^2$ norm.

Forward Euler method giveswhereIts amplification factor isWriting $X={\sin }^2(\vartheta /2)$, the condition $|G{|}^2\le 1$ for every $0\le X\le 1$ is equivalent toThereforewith the second bound omitted when $\alpha =0$.