Crank-Nicolson maximum principle under a step restriction ($\Delta t[(\Delta x{)}^{-2}+(\Delta y{)}^{-2}]\le 1$)

For the two-dimensional heat equation with zero Dirichlet boundary conditions, the Crank-Nicolson diffusion scheme preserves the interval between the minimum and maximum of the initial nodal values and zero under the displayed restriction. The old-time coefficients form a convex combination, while the new-time matrix has positive diagonal, nonpositive off-diagonal entries and row diagonal-dominance margin one. A hypothetical new value above the old maximum makes the new-time row at least that value while its right side is at most the old maximum, a contradiction; apply the reversed inequality at a new minimum. In particular initial nonnegative or nonpositive values preserve their sign. The step restriction guarantees these value bounds even though scalar A-stability holds without it.