Explicit symplectic algorithms based on generating functions for charged particle dynamics

TL;DR

This paper develops explicit symplectic algorithms for charged particle dynamics by combining sum-split and generating function methods, enabling efficient long-term simulations beyond traditional sum-separable Hamiltonian limitations.

Contribution

It introduces a novel approach to construct explicit symplectic algorithms for product-separable Hamiltonians in charged particle dynamics, overcoming previous restrictions.

Findings

Algorithms show improved conservation properties.

Algorithms demonstrate higher efficiency in simulations.

Applicable to complex charged particle systems.

Abstract

Dynamics of a charged particle in the canonical coordinates is a Hamiltonian system, and the well-known symplectic algorithm has been regarded as the de facto method for numerical integration of Hamiltonian systems due to its long-term accuracy and fidelity. For long-term simulations with high efficiency, explicit symplectic algorithms are desirable. However, it is widely accepted that explicit symplectic algorithms are only available for sum-separable Hamiltonians, and that this restriction severely limits the application of explicit symplectic algorithms to charged particle dynamics. To overcome this difficulty, we combine the familiar sum-split method and a generating function method to construct second and third order explicit symplectic algorithms for dynamics of charged particle. The generating function method is designed to generate explicit symplectic algorithms for product-separable Hamiltonian with form of $H(\mathbf{p},\mathbf{q})=\mathbf{p}_{i}f(\mathbf{q})$ or $H(\mathbf{p},\mathbf{q})=\mathbf{q}_{i}f(\mathbf{p})$. Applied to the simulations of charged particle dynamics, the explicit symplectic algorithms based on generating functions demonstrate superiorities in conservation and efficiency.