Exploring Exponential Runge-Kutta Methods: A Survey

TL;DR

This survey comprehensively reviews exponential Runge-Kutta methods, highlighting their historical development, theoretical foundations, and practical applications in numerical integration.

Contribution

It provides an in-depth analysis of the evolution and current state of exponential Runge-Kutta methods, bridging classical and exponential integrators.

Findings

Historical development of exponential Runge-Kutta methods

Examples illustrating the application of these methods

Accessible overview for a broad audience

Abstract

In this survey, we provide an in-depth investigation of exponential Runge-Kutta methods for the numerical integration of initial-value problems. These methods offer a valuable synthesis between classical Runge-Kutta methods, introduced more than a century ago, and exponential integrators, which date back to the 1960s. This manuscript presents both a historical analysis of the development of these methods up to the present day and several examples aimed at making the topic accessible to a broad audience.