Applied Math Seminar: High-order, linearly stable, partitioned solvers for general multiphysics problems based on implicit-explicit Runge-Kutta schemes

Seminar | April 25 | 4-5 p.m. | 740 Evans Hall

 Zhengyu Huang, Stanford University

 Department of Mathematics

The numerical simulation of multiphysics problems is significant in many engineering and scientific applications, e.g., aircraft flutter in transonic flows, biomedical flows in heart and blood vessels, mixing and chemically reacting flows, reactor fuel performance, turbomachinery and so on. These problems are generally highly nonlinear, feature multiple scales and strong coupling effects, and require heterogeneous discretizations for the various physics subsystems. Due to dramatic improvement of single-physics solvers during the last two decades, partitioned procedures for multiphysics system become dominant, which exploit single-physics software components and facilitate mathematical modeling. However, these schemes are often low-order accurate (second order accuracy) and suffer from lack of stability (subiteration is needed).

To relieve these issues, we introduce a general framework for constructing high-order, linearly stable, partitioned solvers for multiphysics problems. The coupled ODE system of the multiphysics problems is taken as a monolithic system and discretized using an implicit-explicit Runge-Kutta (IMEX-RK) discretization based the concept of a predictor for the coupling term. We propose four coupling predictors inspired by basic ideas of weak/strong coupling effects, Jacobi method, and Gauss-Seidel method, which enable the monolithic system to be solved in a partitioned manner, i.e., subsystem-by-subsystem, and preserve the design order of accuracy of the monolithic scheme. We also analyze the stability on a coupled, linear model problem and show that one of the partitioned solvers achieves unconditional linear stability, while the others are unconditionally stable only for certain values of the coupling strength. Furthermore, a fully-discrete adjoint solver derived from our partitioned solvers, is applied for time-dependent PDE constraint optimization. (Joint work with Per-Olof Persson and Matthew J. Zahr)