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

Kun Huang (Corresponding author), Michael Abdelmalik, Boris Breizman, Irene M. Gamba

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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 (pdf), the other for the wave spectral energy density (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 pdf by continuous basis functions, and use discontinuous basis functions for the wave 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 pdf.

Original languageEnglish
Article number112220
Number of pages22
JournalJournal of Computational Physics
Publication statusPublished - 1 Sept 2023


The authors thank and gratefully acknowledge the support from the Oden Institute of Computational Engineering and Sciences and the University of Texas Austin. This project was supported by funding from NSF DMS:2009736 and DOE DE-SC0016283 project Simulation Center for Runaway Electron Avoidance and Mitigation. We thank the reviewers for very valuable comments that help us to improve the present manuscript.

FundersFunder number
Oden Institute of Computational Engineering and Sciences
U.S. Department of EnergyDE-SC0016283
Division of Mathematical Sciences
University of Texas at Austin


    • Computational kinetic systems
    • Mean-field effect
    • Quasilinear theory for magnetized plasmas
    • Weak turbulence model


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