LIRK-W: Linearly-implicit Runge-Kutta methods with approximate matrix factorization

TL;DR

The paper introduces LIRK-W, a new class of linearly implicit Runge-Kutta methods that efficiently handle linear systems with approximate factorizations without splitting the right-hand side.

Contribution

It presents the development of LIRK-W methods, enabling flexible, efficient time integration with arbitrary, time-dependent linear system approximations, and provides their order condition theories.

Findings

LIRK-W methods accommodate arbitrary linear system approximations.

The methods achieve high-order accuracy without splitting the RHS.

Multiple formulations tailored for specific approximation types.

Abstract

This paper develops a new class of linearly implicit time integration schemes called Linearly-Implicit Runge-Kutta-W (LIRK-W) methods. These schemes are based on an implicit-explicit approach which does not require a splitting of the right hand side and allow for arbitrary, time dependent, and stage varying approximations of the linear systems appearing in the method. Several formulations of LIRK-W schemes, each designed for specific approximation types, and their associated order condition theories are presented.