For nonnegative initial data, $U_{\min}=0$. The proved <Crank-Nicolson maximum principle under a step restriction> immediately gives $\boxed{U_{ij}^m\geq0}$ for all indices. For nonpositive initial data, $U_{\max}=0$ and the same bound gives $\boxed{U_{ij}^m\leq0}$. Equivalently, apply positivity to $-U$, since the <finite difference method> is linear and has homogeneous <Dirichlet boundary conditions>. Both conclusions use $\mu_x+\mu_y\leq1$.

Solution

Past exam of the numerical solution of differential equations course of the University of Oxford / 2011 / A / 3 / c / Solution

 Solution (source code)