Additive Operator Splitting Methods for Multiscale and Multiphysics Problems

Shinhoo Kang, Argonne National Laboratory
Shutterstock Earth Model

Many science and engineering problems involve a multiphysics system that combines two or more physics. Earth system models, for example, are composed of various sub-modules such as the global atmosphere, ocean, and land surface. While these components are well developed, integrating them into a single system can be difficult.

Furthermore, each component can have a range of temporal and spatial scales. Computational efficiency, accuracy, and stability are principal concerns. In this talk, we investigate additive operator splitting strategies for tackling these issues. The key idea is to split the governing equations into stiff and nonstiff parts in an additive manner within different time integrators. We first present exponential integrators in the context of discontinuous Galerkin spatial discretization for Burgers and Euler equations. The stiff part is constructed by linearization to extract the fast dynamics of the system. The suggested method not only captures the phase of the fast modes but also is scalable in a modern massively parallel computing architecture.

Next, we introduce implicit-explicit (IMEX) and multirate coupling approaches for a coupled compressible Navier-Stokes equations with an interface condition. IMEX coupling methods explicitly solve one domain while implicitly solving the other domain, whereas multirate coupling methods explicitly solve each partition with different timesteps. In particular, the multirate coupling methods can conserve total mass, have second-order accuracy in time, and give excellent strong- and weak-scaling performance. Finally, we propose entropy-preserving and entropy-stable partitioned Runge-Kutta (RK) methods. We extend the explicit relaxation Runge-Kutta methods to IMEX-RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level.

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