Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 341 / 4 / b / Solution

Write the semidiscrete system as ${\mathbf{u}}^{\prime }=L_h\mathbf{u}+\alpha D_h\mathbf{u}$. With zero boundary values, the centered second-difference matrix $L_h$ is symmetric negative definite and the centered first-difference matrix $D_h$ is skew-symmetric. ThereforeThis is a mesh-uniform stability estimate for the centered convection-diffusion semidiscretization, valid for every real $\alpha$.

Both centered differences have local spatial error $O(h^2)$ for a sufficiently smooth solution. Stability plus consistency gives convergence, equivalently by the semidiscrete form of the Lax equivalence theorem. Thus