High-order partitioned spectral deferred correction solvers for multiphysics problems

TL;DR

This paper introduces a high-order, partitioned spectral deferred correction method for multiphysics problems, enabling the use of existing solvers with improved stability and accuracy for coupled systems.

Contribution

It develops a generic high-order partitioned SDC framework, extending previous IMEX-based methods, with detailed stability analysis and practical demonstrations on various multiphysics problems.

Findings

High-order partitioned SDC schemes are conditionally stable and can be unconditionally stable under certain coupling conditions.

The proposed SDC methods outperform IMEX solvers in robustness for tested multiphysics problems.

Numerical experiments verify the order, stability, and efficiency of the new partitioned SDC solvers.

Abstract

We present an arbitrarily high-order, conditionally stable, partitioned spectral deferred correction (SDC) method for solving multiphysics problems using a sequence of pre-existing single-physics solvers. This method extends the work in [1, 2], which used implicit-explicit Runge-Kutta methods (IMEX) to build high-order, partitioned multiphysics solvers. We consider a generic multiphysics problem modeled as a system of coupled ordinary differential equations (ODEs), coupled through coupling terms that can depend on the state of each subsystem; therefore the method applies to both a semi-discretized system of partial differential equations (PDEs) or problems naturally modeled as coupled systems of ODEs. The sufficient conditions to build arbitrarily high-order partitioned SDC schemes are derived. Based on these conditions, various of partitioned SDC schemes are designed. The stability of the first-order partitioned SDC scheme is analyzed in detail on a coupled, linear model problem. We show that the scheme is conditionally stable, and under conditions on the coupling strength, the scheme can be unconditionally stable. 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, and fluid-structure interaction problems with both incompressible and compressible flows, where we verify the design order of the SDC schemes and study various stability properties. We also directly compare the accuracy, stability, and cost of the proposed partitioned SDC solver with the partitioned IMEX method in [1, 2] on this suite of test problems. The results suggest that the high-order partitioned SDC solvers are more robust than the partitioned IMEX solvers for the numerical examples considered in this work, while the IMEX methods require fewer implicit solves.