 Solution

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.