Poisson solvers for self-consistent multi-particle simulations

TL;DR

This paper reviews efficient numerical methods for solving the Poisson equation in multi-particle simulations, significantly reducing computational complexity from quadratic to linear or near-linear, which enhances the study of beam effects in high-intensity accelerators.

Contribution

It provides a comprehensive review of Poisson solvers that improve computational efficiency in self-consistent multi-particle simulations.

Findings

Numerical methods achieve O(N log N) or O(N) complexity.

Efficient Poisson solvers enable faster simulations of beam effects.

The review aids in selecting suitable methods for high-performance simulations.

Abstract

Self-consistent multi-particle simulation plays an important role in studying beam-beam effects and space charge effects in high-intensity beams. The Poisson equation has to be solved at each time-step based on the particle density distribution in the multi-particle simulation. In this paper, we review a number of numerical methods that can be used to solve the Poisson equation efficiently. The computational complexity of those numerical methods will be O(N log(N)) or O(N) instead of O(N2), where N is the total number of grid points used to solve the Poisson equation.