A conservative Galerkin solver for the quasilinear diffusion model in magnetized plasmas

TL;DR

This paper introduces a conservative Galerkin numerical scheme for the quasilinear diffusion model in magnetized plasmas, ensuring rigorous conservation laws and applicability to various plasma conditions, with demonstrated anisotropic diffusion results.

Contribution

The paper develops a novel, unconditionally conservative Galerkin scheme for the quasilinear plasma model that preserves all conservation laws regardless of transition probabilities.

Findings

The scheme accurately models particle-wave interactions.

Numerical examples show strong anisotropic diffusion effects.

The method is versatile for different plasma regimes.

Abstract

The quasilinear theory describes the resonant interaction between particles and waves with two coupled equations: one for the evolution of the particle probability density function(\textit{pdf}), the other for the wave spectral energy density(\textit{sed}). In this paper, we propose a conservative Galerkin scheme for the quasilinear model in three-dimensional momentum space and three-dimensional spectral space, with cylindrical symmetry. We construct an unconditionally conservative weak form, and propose a discretization that preserves the unconditional conservation property, by "unconditional" we mean that conservation is independent of the singular transition probability. The discrete operators, combined with a consistent quadrature rule, will preserve all the conservation laws rigorously. The technique we propose is quite general: it works for both relativistic and non-relativistic systems, for both magnetized and unmagnetized plasmas, and even for problems with time-dependent dispersion relations. We represent the particle \textit{pdf} by continuous basis functions, and use discontinuous basis functions for the wave \textit{sed}, thus enabling the application of a positivity-preserving technique. The marching simplex algorithm, which was initially designed for computer graphics, is adopted for numerical integration on the resonance manifold. We introduce a semi-implicit time discretization, and discuss the stability condition. In addition, we present numerical examples with a "bump on tail" initial configuration, showing that the particle-wave interaction results in a strong anisotropic diffusion effect on the particle \textit{pdf}.