Efficient energy-preserving exponential integrators for multi-components Hamiltonian systems

TL;DR

This paper introduces energy-preserving exponential integrators for multi-component Hamiltonian systems that are more efficient and stable, especially for highly oscillatory problems, by combining exponential integrators with partitioned averaged vector field methods.

Contribution

The paper develops a novel framework for constructing energy-preserving methods that are computationally more efficient and easier to implement than existing exponential integrators for Hamiltonian systems.

Findings

The proposed methods are more efficient than existing EP-EI.

The schemes are either explicit or linearly implicit, reducing computational cost.

Numerical experiments show improved accuracy and stability for oscillatory problems.

Abstract

In this paper, we develop a framework to construct energy-preserving methods for multi-components Hamiltonian systems, combining the exponential integrator and the partitioned averaged vector field method. This leads to numerical schemes with both advantages of long-time stability and excellent behavior for highly oscillatory or stiff problems. Compared to the existing energy-preserving exponential integrators (EP-EI) in practical implementation, our proposed methods are much efficient which can at least be computed by subsystem instead of handling a nonlinear coupling system at a time. Moreover, for most cases, such as the Klein-Gordon-Schr\"{o}dinger equations and the Klein-Gordon-Zakharov equations considered in this paper, the computational cost can be further reduced. Specifically, one part of the derived schemes is totally explicit, and the other is linearly implicit. In addition, we present rigorous proof of conserving the original energy of Hamiltonian systems, in which an alternative technique is utilized so that no additional assumptions are required, in contrast to the proof strategies used for the existing EP-EI. Numerical experiments are provided to demonstrate the significant advantages in accuracy, computational efficiency, and the ability to capture highly oscillatory solutions.