Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 69 / 3

Use the discrete inner product $⟨U,V{⟩}_h=h{\sum }_{m=1}^{J-1}{\overline{U}}_m{V}_m$, with endpoints $U_0=U_J=0$. The centered first-difference matrix $D_x$ is skew-Hermitian, while summation by parts givesSince $\alpha$ is real, the centered advection contribution has zero real part. ThereforeThis proves the energy contraction for centered drift-diffusion, and hence stability for every fixed real $\alpha$ and every mesh width, in the discrete L2 norm. The extra inequality $|\alpha |h\le 2$ would make the off-diagonal coefficients nonnegative and support a discrete maximum principle, but is not needed for this energy proof. No time-stepping method is involved here.

For a sufficiently smooth exact solution, the centered spatial residual isLet $e=U-R_{h}u$ be the nodal error. It satisfies $e^{\prime }=A_{h}e+{\tau }_h$, with $A_h=D_{xx}-\alpha D_x$ and $\Vert {\tau }_h{\Vert }_h\le C{h}^2$ on the fixed time interval. Part (a) gives $\Vert e^{t{A}_h}\Vert \le 1$, so Duhamel's formula yieldsThus the method is convergent, with second-order spatial error for smooth data and second-order compatible initialization. The homogeneous endpoint values introduce no boundary residual. For nonsmooth L2 data, stable approximating projections and a density argument extend numerical convergence, although the same order need not hold without the derivative bounds used above.