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

Abstract

The numerical solution of coupled differential equation systems is usually done following a monolithic or a decoupled algorithm. In contrast to the holistic monolithic solvers, the decoupled solution strategies are based on breaking down the system into several subsystems. This results in different characteristics of these families of solvers, e. g., while the monolithic algorithms provide a relatively straight-forward solution framework, unlike their decoupled counterparts, they hinder software re-usability and customisation. This is a drawback for multi-field and multi-rate problems. The reason is that a multi-field problem comprises several subproblems corresponding to interacting subsystems. This suggests exploiting an individual solver for each subproblem. Moreover, for the efficient solution of a multi-rate problem, it makes sense to perform the temporal integration of each subproblem using a time-step size relative to its evolution rate. Nevertheless, decoupled solvers introduce additional errors to the solution and, thus, they must always be accompanied by a thorough stability analysis.Here, tailored solution schemes for the decoupled solution of multi-field and multi-rate problems are proposed. Moreover, the stability behaviour of the solutions obtained from these methods are studied. Numerical examples are solved and the reliability of the outcome of the stability analysis is investigated.

Multi-rate time stepping schemes for hydro-geomechanical model for subsurface methane hydrate reservoirs

“…The application of MRT methods, especially the Implicit-Explicit methods (IMEX), is becoming increasingly popular in the PDE community. Some of the recent extensions of these methods to application areas of coupled free and porous media flows, air pollution modelling, multi-scale fluid-solid interaction, among others, can be found in [17,21,24,25,26].…”