On the stability of ADI methods

  • Ole Østerby DEPARTMENT OF COMPUTER SCIENCE AARHUS UNIVERSITY IT-parken, Aabogade 34 DK-8200 Aarhus N, Denmark

Abstract

When solving parabolic equations in two space dimensions implicit methods are preferred to the explicit method because of their better stability properties. Straightforward implementation of implicit methods require time-consuming solution of large systems of linear equations, and ADI methods are preferred instead. We expect the ADI methods to inherit the stability properties of the implicit methods they are derived from, and we demonstrate that this is partly true. The Douglas-Rachford and Peaceman-Rachford methods are absolutely stable in the sense that their growth factors are ≤ 1 in absolute value. Near jump discontinuities, however, there are differences w.r.t. how the ADI methods react to the situation: do they produce oscillations and how effectively do they damp them. We demonstrate the behaviour on two simple examples.

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Published
2016-04-10
How to Cite
Østerby, O. (2016). On the stability of ADI methods. DAIMI Report Series, 42(598). https://doi.org/10.7146/dpb.v42i598.25146