Structure-preserving techniques in accelerator physics

TL;DR

This paper discusses advanced structure-preserving numerical techniques for accurately simulating long-term particle trajectories in accelerators, emphasizing symplectic methods to maintain physical fidelity over billions of revolutions.

Contribution

It introduces methods for constructing high-order symplectic maps tailored for realistic magnetic field models in accelerator physics.

Findings

High-order symplectic maps improve long-term stability predictions.

Traditional numerical methods struggle with long-term accuracy in accelerator simulations.

Symplectic condition preservation is crucial for realistic modeling.

Abstract

To a very good approximation, particularly for hadron machines, charged-particle trajectories in accelerators obey Hamiltonian mechanics. During routine storage times of eight hours or more, such particles execute some $10^{8}$ revolutions about the machine, $10^{10}$ oscillations about the design orbit, and $10^{13}$ passages through various bending and focusing elements. Prior to building, or modifying, such a machine, we seek to identify accurately the long-term behavior and stability of particle orbits over such large numbers of interactions. This demanding computational effort does not yield easily to traditional methods of symplectic numerical integration, including both explicit Yoshida-type and implicit Runge-Kutta or Gaussian methods. As an alternative, one may compute an approximate one-turn map and then iterate that map. We describe some of the essential considerations and techniques for constructing such maps to high order and for realistic magnetic field models. Particular attention is given to preserving the symplectic condition characteristic of Hamiltonian mechanics.