Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 80 / 2 / b

Write $h=\Delta x$, impose $u_0=u_{M+1}=0$, and use the discrete inner product $⟨u,v{⟩}_h=h{\sum }_{m=1}^M{\overline{u}}_m{v}_m$. Discrete integration by parts for the second difference givesFor the centered drift, reindexing with the zero endpoints givessince the two sums are complex conjugates. This is the fact that centered discrete advection is skew-adjoint. The semidiscrete energy estimate is thereforeThe finite-dimensional linear system has a unique solution for all times, and $\Vert u(t){\Vert }_h\le \Vert u(0){\Vert }_h$, uniformly in the mesh and in real $\kappa$. Differences of two semidiscrete solutions obey the same bound, which is the required stability.

One may also use the smallest eigenvalue ${\lambda }_h=4h^{-2}{\sin }^2(\pi /[2(M+1)])$ of the negative second-difference operator to obtain $\Vert u(t){\Vert }_h\le e^{-{\lambda }_{h}t}\Vert u(0){\Vert }_h$. This conclusion concerns the method of lines; a later choice of time integrator has its own stability conditions.