Application of approximate matrix factorization to high order linearly implicit Runge-Kutta methods

TL;DR

This paper explores the use of approximate matrix factorization in high order linearly implicit Runge-Kutta methods, introducing a correction procedure to maintain accuracy and stability for large, stiff systems.

Contribution

It presents a novel correction technique that preserves high order accuracy in approximate factorization-based Runge-Kutta methods for stiff differential equations.

Findings

The correction procedure effectively retains high order accuracy.

The methods demonstrate stability and efficiency in reaction-diffusion problems.

Numerical experiments confirm the approach's suitability for large, stiff systems.

Abstract

Linearly implicit Runge-Kutta methods with approximate matrix factorization can solve efficiently large systems of differential equations that have a stiff linear part, e.g. reaction-diffusion systems. However, the use of approximate factorization usually leads to loss of accuracy, which makes it attractive only for low order time integration schemes. This paper discusses the application of approximate matrix factorization with high order methods; an inexpensive correction procedure applied to each stage allows to retain the high order of the underlying linearly implicit Runge-Kutta scheme. The accuracy and stability of the methods are studied. Numerical experiments on reaction-diffusion type problems of different sizes and with different degrees of stiffness illustrate the efficiency of the proposed approach.