A Spectral Canonical Electrostatic Algorithm

TL;DR

This paper introduces a gridless spectral electrostatic algorithm based on Hamilton's Principle that conserves energy and momentum, avoiding unphysical grid heating in long-term particle simulations.

Contribution

The paper develops a novel spectral electrostatic algorithm derived from Hamilton's Principle, eliminating grid heating and preserving physical invariants in particle simulations.

Findings

The algorithm conserves energy and momentum in two-body problems.

It avoids unphysical grid heating common in particle-in-cell methods.

Demonstrates stability over many oscillation periods.

Abstract

Studying single-particle dynamics over many periods of oscillations is a well-understood problem solved using symplectic integration. Such integration schemes derive their update sequence from an approximate Hamiltonian, guaranteeing that the geometric structure of the underlying problem is preserved. Simulating a self-consistent system over many oscillations can introduce numerical artifacts such as grid heating. This unphysical heating stems from using non-symplectic methods on Hamiltonian systems. With this guidance, we derive an electrostatic algorithm using a discrete form of Hamilton's Principle. The resulting algorithm, a gridless spectral electrostatic macroparticle model, does not exhibit the unphysical heating typical of most particle-in-cell methods. We present results of this using a two-body problem as an example of the algorithm's energy- and momentum-conserving properties.