On the Stability Analysis of Decoupled Solution Schemes

Abstract

AbstractMany problems in engineering, physics or other disciplines require an integrated treatment of coupled fields. These problems are characterised by a dynamic interaction among two or more physically or computationally distinct components, where the undergoing mathematical model commonly consists of a system of coupled PDE. Considerable progress has been made in the development of appropriate computational schemes to solve such coupled PDE systems. These attempts have resulted in various monolithic and decoupled numerical solution approaches. Despite the unconditional stability offered by implicit monolithic solution strategies, their use is not always recommended. The reason mainly lies in the complexity of the resulting system of equations and the limited flexibility in choosing appropriate time integrators for individual components. This has motivated the elaboration of tailored decoupled solution schemes, which follow the idea of splitting the problem into several sub‐problems. But selection of the way of splitting can have a direct influence on the stability of the resulting solution algorithm. This necessitates the stability analysis of such an algorithm.Here, we introduce a general framework for the stability analysis of decoupled solution schemes. The approach is then used to study the stability behaviour of established decoupling strategies applied to typical volume‐ and surface‐coupled problems, namely, coupled problems of thermoelasticity, porous media dynamics and structure‐structure interaction. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)