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

Take the real $L^2$ inner product of the equation with $u$. The homogeneous Dirichlet boundary conditions and integration by parts giveThus $\Vert u(t){\Vert }_{L^2}\le \Vert u_0{\Vert }_{L^2}$, and applying the same estimate to the difference of two solutions gives continuous dependence on the initial data. The constant-advection term is skew-symmetric under these boundary conditions and contributes no energy. Hence

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