Simulating Pitch Angle Scattering Using An Explicitly Solvable Energy Conserving Algorithm

TL;DR

This paper introduces a new explicit, energy-conserving numerical algorithm for simulating pitch angle scattering in plasma physics, ensuring physical accuracy and stability over long simulations.

Contribution

The paper presents a novel structure-preserving, explicitly solvable algorithm for Langevin equations in pitch angle scattering, maintaining energy conservation and high accuracy.

Findings

The algorithm preserves velocity norms to machine precision.

It has the same convergence order as Euler-Maruyama.

Benchmark tests show excellent agreement with analytical solutions.

Abstract

Particle distribution functions evolving under the Lorentz operator can be simulated with the Langevin equation for pitch angle scattering. This approach is frequently used in particle based Monte-Carlo simulations of plasma collisions, among others. However, most numerical treatments do not guarantee energy conservation, which may lead to unphysical artifacts such as numerical heating and spectra distortions. We present a novel structure-preserving numerical algorithm for the Langevin equation for pitch angle scattering. Similar to the well-known Boris algorithm, the proposed numerical scheme takes advantage of the structure-preserving properties of the Cayley transform when calculating the velocity-space rotations. The resulting algorithm is explicitly solvable, while preserving the norm of velocities down to machine precision. We demonstrate that the method has the same order of numerical convergence as the traditional stochastic Euler-Maruyama method. The numerical scheme is benchmarked by simulating the pitch angle scattering of a particle beam, and comparing with the analytical solution. Benchmark results show excellent agreement with theoretical predictions, showcasing the remarkable long time accuracy of the proposed algorithm.