W-Methods and Approximate Matrix Factorization for Parabolic PDEs with Mixed Derivative Terms

In this chapter W-methods are combined with the Approximate Matrix Factorization technique (AMF) in alternating direction implicit (ADI) sense for the time integration of parabolic partial differential equations with mixed derivatives in the elliptic operator, previously discretized in space by means of Finite Differences. Three different families of AMF-type W-methods are introduced and their unconditional stability is analized regardless of the spatial dimension. To this aim, a scalar test equation is presented and it is shown to be relevant for the class of problems under consideration when either periodic or homogeneous Dirichlet boundary conditions are imposed. Numerical results comparing the proposed AMF-type W-methods and some classical ADI schemes in the literature for 2 ≤ m ≤ 4 space dimensions are presented.