Two new classes of exponential Runge–Kutta integrators for efficiently solving stiff systems or highly oscillatory problems

Abstract

We note a fact that stiff systems or differential equations that have highly oscillatory solutions cannot be solved efficiently using conventional methods. In this paper, we study two new classes of exponential Runge–Kutta (ERK) integrators for efficiently solving stiff systems or highly oscillatory problems. We first present a novel class of explicit modified version of exponential Runge–Kutta (MVERK) methods based on the order conditions. Furthermore, we consider a class of explicit simplified version of exponential Runge–Kutta (SVERK) methods. Numerical results demonstrate the high efficiency of the explicit MVERK integrators and SVERK methods derived in this paper compared with the well-known explicit ERK integrators for stiff systems or highly oscillatory problems in the literature.

Keywords:

• Exponential integrators
• stiff systems
• highly oscillatory problems
• explicit methods
• Runge–Kutta-type methods

2010 Mathematics Subject Classifications:

• 65L04
• 65L05
• 65L06
• 65L08

Acknowledgments

We sincerely thank the anonymous reviewers for their valuable suggestions, which helped improve the expression of this manuscript.This work was supported by NSFC (12371403).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 It is noted that for this γ and the α introduced in (37(37) $0\le \alpha <1$(37) ), this is no relation between them.