= Solution
Choose the centered <Dirichlet discrete Laplacian> in each direction. With $T=\operatorname{tridiag}(1,-2,1)$ put $A=T\otimes I$, $B=I\otimes T$ and $\mu=\Delta t/h^2$. Thus $h^{-2}A$ and $h^{-2}B$ are the physical derivative approximations; the factor $h^{-2}$ has been absorbed into the <Courant number> in the written update. Equivalently use the scaled derivative <matrices> and the time parameter $\Delta t$.
For a one-dimensional vector with endpoint values $v_0=v_{m+1}=0$,
$$
v^TTv=-\sum_{j=0}^m(v_{j+1}-v_j)^2\leq0.
$$
Thus $A$ and $B$ are real symmetric negative-definite <matrices>. An orthonormal <eigenvector> basis shows that all <eigenvalues> of each exponential lie in $(0,1]$ for $\mu\geq0$, so
$$
\|E(\mu;A,B)\|_2
\leq\|e^{\mu A/2}\|_2^2\|e^{\mu B}\|_2\leq1.
$$
Iterating gives \b[$\|U^n\|_2\leq\|U^0\|_2$ for every nonnegative step parameter], a mesh-independent unconditional <stability> bound. Multiplying the squared norm by the grid-area weight $h^2$ leaves the same estimate. In this natural tensor-grid choice $AB=BA=T\otimes T$, so $E(\mu;A,B)=e^{\mu(A+B)}$: the split update is even the exact semidiscrete diffusion flow. Contractivity of a product of these negative symmetric exponentials would prove <stability> even without commutation.
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