New explicit finite difference schemes for two-dimensional diffusion subject to specification of mass

Abstract

Two different explicit finite difference schemes for the numerical solution of the diffusion equation on a rectangular region, subject to local or nonlocal boundary conditions, the latter involving a double integral to simulate specification of mass in a curved region, are compared. These schemes, the two-dimensional 9-point Forward Time Centered Space (FTCS) explicit formula [Noye & Hayman, J Comp Math 42, 1992, 223–236] and the locally one-dimensional (LOD) method based on the classical one-dimensional FTCS formula [Noye & Hayman, J Comp Math 51, 1994, 215–228], are economical to use, are generally second-order, have bounded ranges of stability, and can be shown to be identical at grid points in the interior of the solution domain. However, results obtained are different, unless a special boundary treatment is used with the LOD method. Then the LOD method is more efficient. Some numerical tests are presented for both cases, and accuracy and Central Processor (CP) time needed for the nonlocal problem are found to be superior than those for the method of Cannon et al. [Cannon et al., Appl Anal J 50, 1993, 1–19]. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 521–534, 1999