Stability of ADI schemes for multidimensional diffusion equations with mixed derivative terms

TL;DR

This paper analyzes the unconditional stability of four ADI schemes for multidimensional diffusion equations with mixed derivatives, providing new stability conditions that depend on the mixed derivative coefficients.

Contribution

It generalizes previous stability results by deriving necessary and sufficient conditions on scheme parameters considering mixed derivative coefficients.

Findings

Derived stability conditions for ADI schemes with mixed derivatives

Validated theoretical results through numerical experiments

Extended previous stability analyses to more general cases

Abstract

In this paper the unconditional stability of four well-known ADI schemes is analyzed in the application to time-dependent multidimensional diffusion equations with mixed derivative terms. Necessary and sufficient conditions on the parameter theta of each scheme are obtained that take into account the actual size of the mixed derivative coefficients. Our results generalize results obtained previously by Craig & Sneyd (1988) and In 't Hout & Welfert (2009). Numerical experiments are presented illustrating our main theorems.