Explicit finite difference methods for two-dimensional diffusion with a non-local boundary condition

Two different explicit finite difference methods for solving the heat equation in two-dimensional space with non-local boundary conditions are considered. The schemes are based on the standard explicit second-order FTCS method and the Noye and Hayman explicit method [1] which is generally second order, but is fourth order when the diffusion number takes the value s = 16 and is stable for much larger diffusion numbers than the former. The CPU time needed for each of the methods is also considered and are compared with those of the implicit method developed by Cannon et al. [2]. Error estimates derived in the maximum norm are tabulated.