Simplified explicit exponential Runge-Kutta methods without order reduction

TL;DR

This paper introduces simplified explicit exponential Runge-Kutta methods that avoid order reduction in solving nonlinear initial boundary value problems, eliminating the need for numerical differentiation and simplifying boundary data handling.

Contribution

It significantly simplifies the discretization formulas for explicit exponential Runge-Kutta methods, making them more practical and computationally efficient for boundary value problems.

Findings

Simplified formulas for stages and solutions using fewer $\

The new methods avoid order reduction without numerical differentiation.

They are computationally competitive with existing high-order methods.

Abstract

In a previous paper, a technique was suggested to avoid order reduction with any explicit exponential Runge-Kutta method when integrating initial boundary value nonlinear problems with time-dependent boundary conditions. In this paper, we significantly simplify the full discretization formulas to be applied under conditions which are nearly always satisfied in practice. Not only a simpler linear combination of $\varphi_j$-functions is given for both the stages and the solution, but also the information required on the boundary is so much simplified that, in order to get local order three, it is no longer necessary to resort to numerical differentiation in space. The technique is then shown to be computationally competitive against other widely used methods with high enough stiff order through the standard method of lines.