Eliminating Order Reduction on Linear, Time-Dependent ODEs with GARK Methods

TL;DR

This paper introduces a flexible augmentation technique for Runge-Kutta methods that prevents order reduction when solving stiff, linear, time-dependent ODEs, ensuring methods achieve their theoretical convergence rates.

Contribution

It proposes a novel augmentation approach within the additive Runge-Kutta framework to eliminate order reduction in linear, time-dependent ODEs, applicable to various Runge-Kutta schemes.

Findings

Eliminates order reduction for tested Runge-Kutta methods.

Achieves theoretical convergence orders in numerical experiments.

Applicable to explicit, diagonally implicit, and fully implicit schemes.

Abstract

When applied to stiff, linear differential equations with time-dependent forcing, Runge-Kutta methods can exhibit convergence rates lower than predicted by the classical order condition theory. Commonly, this order reduction phenomenon is addressed by using an expensive, fully implicit Runge-Kutta method with high stage order or a specialized scheme satisfying additional order conditions. This work develops a flexible approach of augmenting an arbitrary Runge-Kutta method with a fully implicit method used to treat the forcing such as to maintain the classical order of the base scheme. Our methods and analyses are based on the general-structure additive Runge-Kutta framework. Numerical experiments using diagonally implicit, fully implicit, and even explicit Runge-Kutta methods confirm that the new approach eliminates order reduction for the class of problems under consideration, and the base methods achieve their theoretical orders of convergence.