Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 341 / 3

Write $h=\Delta x$ and set $u_0=u_{M+1}=0$. In the discrete L2 norm $\Vert u{\Vert }_h^2=h{\sum }_{m=1}^M|u_m{|}^2$, the energy method givesTo obtain this identity, shift the indices in the centered second-difference sum. The centered first-difference matrix is skew-symmetric, so its contribution has zero real part even for complex grid values. The homogeneous Dirichlet boundary conditions remove the endpoint terms.

Therefore the semidiscretization is stable for every real $\alpha$ and every positive $h$, with the mesh-independent bound $\Vert u(t){\Vert }_h\le \Vert u(0){\Vert }_h$. No restriction on $|\alpha |h$ is needed for this energy estimate.

With $h=\Delta x$, $\tau =\Delta t$, $r=\tau /h^2$ and $c=\alpha \tau /h$, the Forward Euler method givesFor a Fourier mode $e^{im\theta }$, the amplification factor is $G(\theta )=1-4r{\sin }^2(\theta /2)+ic\sin \theta$. Set $s={\sin }^2(\theta /2)$. ThenThe expression in braces is an affine function of $s$, so it is nonpositive for all $s\in [0,1]$ precisely when its two endpoint values are nonpositive. For $\tau >0$, these conditions are $c^2\le 2r$ and $r\le 1/2$. Thus Forward Euler stability for centered advection-diffusion requires exactlyFor $\alpha =0$, only the first condition remains. If either bound fails, a band of Fourier modes has $|G|>1$, giving instability under von Neumann stability analysis. At equality the amplification bound still holds, so the endpoints are included. This is discrete L2 norm stability; nonnegative stencil weights would impose the additional spatial restriction $|\alpha |\Delta x\le 2$.