Fully implicit finite differences methods for two-dimensional diffusion with a non-local boundary condition

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Three new fully implicit methods which are based on the (5,5) Crank-Nicolson method, the (5,5) N-H (Noye-Hayman) implicit method and the (9,9) N-H implicit method are developed for solving the heat equation in two dimensional space with non-local boundary conditions. The latter is fourth-order while the others are second-order. While the implicit methods developed here, like the scheme based on the standard implicit backward time centered space (BTCS) method, use a large amount of central processor (CPU) time, the high accuracy of the new fourth-order fully implicit scheme is significant. Like the BTCS method, the new methods are also unconditionally stable.