High-order, linearly stable, partitioned solvers for general multiphysics problems based on implicit–explicit Runge–Kutta schemes

• Published in: Computer methods in applied mechanics and engineering 2019-04, Vol.346 (C), p.674-706
• Main Authors: Huang, D.Z., Persson, P.-O., Zahr, M.J.
• Format: Article
• Language: English
• Subjects: Coupling; Discretization; Fluid–structure interaction; High-order methods; Implicit–explicit Runge–Kutta; Mathematical models; MATHEMATICS AND COMPUTING; Method of lines; Multiphysics simulations; Ordinary differential equations; Partial differential equations; Partitioned solvers; Runge-Kutta method; Software reuse; Solvers; Stability analysis; Subsystems
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2018.09.015

This work introduces a general framework for constructing high-order, linearly stable, partitioned solvers for multiphysics problems from a monolithic implicit–explicit Runge–Kutta (IMEX-RK) discretization of the semi-discrete equations. The generic multiphysics problem is modeled as a system of n systems of partial differential equations where the ith subsystem is coupled to the other subsystems through a coupling term that can depend on the state of all the other subsystems. This coupled system of partial differential equations reduces to a coupled system of ordinary differential equations via the method of lines where an appropriate spatial discretization is applied to each subsystem. The coupled system of ordinary differential equations is taken as a monolithic system and discretized using an IMEX-RK discretization with a specific implicit–explicit decomposition that introduces the concept of a predictor for the coupling term. We propose four coupling predictors that enable the monolithic system to be solved in a partitioned manner, i.e., subsystem-by-subsystem, and preserve the IMEX-RK structure and therefore the design order of accuracy of the monolithic scheme. The four partitioned solvers that result from these predictors are high-order accurate, allow for maximum re-use of existing single-physics software, and two of the four solvers allow the subsystems to be solved in parallel at a given stage and time step. We also analyze the stability of a coupled, linear model problem with a specific coupling structure and show that one of the partitioned solvers achieves unconditional linear stability for this problem, while the others are unconditionally stable only for certain values of the coupling strength. We demonstrate the performance of the proposed partitioned solvers on several classes of multiphysics problems including a simple linear system of ODEs, advection–diffusion–reaction systems, fluid–structure interaction problems, and particle-laden flows, where we verify the design order of the IMEX schemes and study various stability properties.