Pseudo-Energy-Preserving Explicit Runge-Kutta Methods

TL;DR

This paper develops explicit Runge-Kutta methods that approximately preserve energy in Hamiltonian systems up to a certain order, providing new high-order methods that outperform classical ones in energy conservation over moderate times.

Contribution

It introduces the concept of pseudo-energy-preserving (PEP) order for explicit Runge-Kutta methods and constructs methods up to PEP order six, demonstrating their effectiveness in energy conservation.

Findings

PEP methods behave like energy-conservative methods over moderate times

PEP methods have smaller errors than classical methods of the same order

Constructed methods achieve PEP order up to six

Abstract

Using a recent characterization of energy-preserving B-series, we derive the explicit conditions on the coefficients of a Runge-Kutta method that ensure energy preservation (for Hamiltonian systems) up to a given order in the step size, which we refer to as the pseudo-energy-preserving (PEP) order. We study explicit Runge-Kutta methods with PEP order higher than their classical order. We provide examples of such methods up to PEP order six, and test them on Hamiltonian ODE and PDE systems. We find that these methods behave similarly to exactly energy-conservative methods over moderate time intervals and exhibit significantly smaller errors, relative to other Runge-Kutta methods of the same order, for moderately long-time simulations.