 Solution

Use the Fourier amplification symbol for the semidiscrete generator. At a mode $e^{i(k\xi +l\eta )}$, the two stencil symbols areThe mass symbol satisfies $1/3\le m\le 1$. The numerator of $\ell$ is increasing in each cosine, since each partial derivative is at least two, and its maximum is zero at $(1,1)$. Hence $\ell \le 0$ and the generator ${\lambda }_h=\ell /m$ is real and nonpositive.

Every exact-time mode is multiplied by $e^{t{\lambda }_h}$, of modulus at most one. Parseval identity therefore provesThe method is stable, uniformly in the spatial mesh. This is a semidiscrete conclusion: it imposes no time-step condition until a particular time integrator is applied. For the stated merely square-integrable initial data, cell averages give a well-defined stable initialization, with $\Vert U(0){\Vert }_{{\ell }_h^2}\le \Vert v{\Vert }_{L^2}$ by Cauchy–Schwarz. Arbitrary point sampling of an L2 equivalence class is not required.