Stability Analysis of Decoupled Solution Strategies for Coupled Multi‐field Problems – A General Framework

Abstract

A coupled partial differential equation (PDE) system, stemming from the mathematical modelling of a coupled phenomenon, is usually solved numerically following a monolithic or a decoupled solution method. In spite of the potential unconditional stability offered by monolithic solvers, their usage for solving complex problems sometimes proves cumbersome. This has motivated the development of various partitioned and staggered solution strategies, generally known as decoupled solution schemes. To this end, the problem is broken down into several isolated yet communicating sub-problems that are independently advanced in time, possibly by different integrators. Nevertheless, using a decoupled solver introduces additional errors to the system and, therefore, may jeopardise the stability of the solution [1]. Consequently, to scrutinise the stability of the solution scheme becomes a pertinent step in proposing decoupled solution strategies. Here, we endeavour to present a practical stability analysis algorithm, which can readily be used to reveal the stability condition of numerical solvers. To illustrate its capabilities, the algorithm is then utilised for the stability analysis of solution schemes applied to multi variate coupled PDE systems resulting from the mathematical modelling of surface-and volume-coupled multi-field problems.

Coupled Multi‐field and Multi‐rate Problems – Numerical Solution and Stability Analysis

“…The consistency of the critical time-step sizes calculated from this relation with what was obtained from the numerical solution of 1-and 2-d examples has already been presented [2,5]. To complete the investigation, two 3-d initial-boundary-value problems as depicted in Figs.…”