Explicit exponential Runge-Kutta methods of high order for parabolic problems

TL;DR

This paper develops a new fifth-order explicit exponential Runge-Kutta method for semilinear parabolic problems, overcoming previous order limitations with a novel approach to stiff order conditions and demonstrating its effectiveness through convergence analysis and numerical tests.

Contribution

It introduces a novel approach to derive stiff order conditions, enabling the construction of the first fifth-order explicit exponential Runge-Kutta method.

Findings

Constructed a fifth-order method with 8 stages.

Proved convergence for semilinear parabolic problems.

Numerical example confirms theoretical convergence bound.

Abstract

Exponential Runge-Kutta methods constitute efficient integrators for semilinear stiff problems. So far, however, explicit exponential Runge-Kutta methods are available in the literature up to order 4 only. The aim of this paper is to construct a fifth-order method. For this purpose, we make use of a novel approach to derive the stiff order conditions for high-order exponential methods. This allows us to obtain the conditions for a method of order 5 in an elegant way. After stating the conditions, we first show that there does not exist an explicit exponential Runge-Kutta method of order 5 with less than or equal to 6 stages. Then, we construct a fifth-order method with 8 stages and prove its convergence for semilinear parabolic problems. Finally, a numerical example is given that illustrates our convergence bound.