Linearly implicit GARK schemes

TL;DR

This paper introduces new multimethod GARK schemes based on linearly implicit Rosenbrock methods, providing a flexible framework for efficiently solving multiphysics systems with multiple time scales and stiffness levels.

Contribution

It develops a general order condition theory for linearly implicit GARK methods, including for differential-algebraic equations, and constructs practical schemes up to order four.

Findings

Developed GARK-ROS/GARK-ROW families of multimethods

Established order conditions for linearly implicit methods

Constructed practical schemes up to order four

Abstract

Systems driven by multiple physical processes are central to many areas of science and engineering. Time discretization of multiphysics systems is challenging, since different processes have different levels of stiffness and characteristic time scales. The multimethod approach discretizes each physical process with an appropriate numerical method; the methods are coupled appropriately such that the overall solution has the desired accuracy and stability properties. The authors developed the general-structure additive Runge-Kutta (GARK) framework, which constructs multimethods based on Runge-Kutta schemes. This paper constructs the new GARK-ROS/GARK-ROW families of multimethods based on linearly implicit Rosenbrock/Rosenbrock-W schemes. For ordinary differential equation models, we develop a general order condition theory for linearly implicit methods with any number of partitions, using exact or approximate Jacobians. We generalize the order condition theory to two-way partitioned index-1 differential-algebraic equations. Applications of the framework include decoupled linearly implicit, linearly implicit/explicit, and linearly implicit/implicit methods. Practical GARK-ROS and GARK-ROW schemes of order up to four are constructed.