An Explicitly Solvable Energy-Conserving Algorithm for Pitch-Angle Scattering in Magnetized Plasmas

TL;DR

This paper introduces an explicit, energy-conserving numerical algorithm for simulating pitch-angle scattering in magnetized plasmas, overcoming convergence issues of standard methods and validated by analytical benchmarks.

Contribution

The paper presents a novel explicitly solvable energy-conserving algorithm for SDEs in plasma physics, addressing divergence issues and proving strong convergence.

Findings

The ESEC algorithm is order 1/2 strongly convergent.

Numerical results agree well with analytical solutions.

The method effectively conserves energy in simulations.

Abstract

We develop an Explicitly Solvable Energy-Conserving (ESEC) algorithm for the Stochastic Differential Equation (SDE) describing the pitch-angle scattering process in magnetized plasmas. The Cayley transform is used to calculate both the deterministic gyromotion and stochastic scattering, affording the algorithm to be explicitly solvable and exactly energy conserving. An unusual property of the SDE for pitch-angle scattering is that its coefficients diverge at the zero velocity and do not satisfy the global Lipschitz condition. Consequently, when standard numerical methods, such as the Euler-Maruyama (EM), are applied, numerical convergence is difficult to establish. For the proposed ESEC algorithm, its energy-preserving property enables us to overcome this obstacle. We rigorously prove that the ESEC algorithm is order 1/2 strongly convergent. This result is confirmed by detailed numerical studies. For the case of pitch-angle scattering in a magnetized plasma with a constant magnetic field, the numerical solution is benchmarked against the analytical solution, and excellent agreements are found.