New lod and adi methods for the two-dimensional diffusion equation

Abstract

This paper extends earlier work on the solution of the constant-coefficient two-dimensional diffusion equation by considering two classes of time-split finite-difference methods, namely locally one-dimensional (LOD) schemes and alternating direction implicit (ADI) methods. Two new fourth-order techniques are described and tested. Firstly, a LOD method based on the fourth-order explicit Noye-Hayman procedure for the one-dimensional diffusion equation is described. Proper treatment of values at, and adjacent to, the boundary at intermediate time levels is necessary, otherwise the method degenerates to second-order. Secondly, an unconditionally stable ADI method based on a (3,9) two-dimensional computational molecule is developed and tested.