A unified formulation of splitting-based implicit time integration schemes

TL;DR

This paper introduces a unified framework for splitting-based implicit time integration schemes using GARK methods, enabling systematic analysis and development of new methods for partitioned differential equations.

Contribution

It develops the IMIM GARK scheme class, provides order and stability conditions, and unifies classical splitting methods within this framework.

Findings

Classical splitting methods are special cases of IMIM GARK schemes.

New IMIM-GARK splitting methods are successfully tested on parabolic systems.

The framework facilitates analysis and design of efficient partitioned integrators.

Abstract

Splitting-based time integration approaches such as fractional steps, alternating direction implicit, operator splitting, and locally one-dimensional methods partition the system of interest into components and solve individual components implicitly in a cost-effective way. This work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge-Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions and stability analysis for this class, and explain their application to partitioned systems of ordinary differential equations. We show that classical splitting methods belong to the IMIM GARK family, and therefore can be studied in this unified framework. New IMIM-GARK splitting methods are developed and tested using parabolic systems.