Implicit locally one-dimensional methods for two-dimensional diffusion with a non-local boundary condition

Two new second-order finite difference techniques based upon the classical 3-point backward time centered space (BTCS) method and the Crank–Nicolson scheme, and also a fourth-order finite difference scheme based on Crandall's method for one-dimensional diffusion, are used to solve the two-dimensional time dependent diffusion equation with non-local boundary conditions. In these cases locally one-dimensional (LOD) techniques are used to extend the one-dimensional techniques to solve the two-dimensional problem. The stability properties and truncation error of these methods are discussed and the results of a numerical experiment for these three methods are presented. Error estimates are also tabulated. The results of numerical testing shows that these schemes uses less central processor (CPU) time than the fully implicit schemes.