= Solution
For an error <Fourier mode> $e_j^n=\widehat e^n e^{i\theta j}$, the periodic <discrete Laplacian> has eigenvalue $-4\sin^2(\theta/2)/h^2$. The <Backward Euler method> therefore has amplification factor
$$
\boxed{g(\theta)=\frac1{1+4r\sin^2(\theta/2)},\qquad0<g\le1\quad(r\ge0).}
$$
<von 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.
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