 Solution

For an error Fourier mode $e_j^n={\hat{e}}^n{e}^{i\theta j}$, the periodic discrete Laplacian has eigenvalue $-4{\sin }^2(\theta /2)/h^2$. The Backward Euler method therefore has amplification factorvon Neumann stability analysis thus imposes no upper limit on positive $\Delta \tau$. The constant mode has $g=1$ and is not damped; compatible forcing and a gauge condition remain necessary as discussed in part (e). Large steps can accelerate relaxation of nonconstant modes, but require a suitably accurate linear solver and can worsen its conditioning. Unconditional linear stability is not a statement about time accuracy or cost.