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A structure-preserving multiscale solver for particle-wave interaction in non-uniform magnetized plasmas

Kun Huang, Irene M. Gamba, Chi-Wang Shu

TL;DR

This work introduces a structure-preserving multiscale solver for particle–plasmon interactions in non-uniform magnetized plasmas, combining a conservative LDG discretization for diffusion–interaction terms with a trajectory-averaging approach for the fast plasmon Hamiltonian flow. It leverages trajectory bundles and a connection-proportion algorithm to discretize the kernel of the advection operator, ensuring exact mass, momentum, and energy conservation in the averaged system. The methodology yields substantial computational savings while maintaining fidelity to the underlying physics, as demonstrated by numerical tests in a non-uniform cylinder geometry that show position-dependent diffusion and strict conservation. The framework lays groundwork for large-scale runaway electron simulations and can be extended to more complex quasilinear models and geometries.

Abstract

Particle-wave interaction is of fundamental interest in plasma physics, especially in the study of runaway electrons in magnetic confinement fusion. Analogous to the concept of photons and phonons, wave packets in plasma can also be treated as quasi-particles, called plasmons. To model the ``mixture" of electrons and plasmons in plasma, a set of ``collisional" kinetic equations has been derived, based on weak turbulence limit and the Wentzel-Kramers-Brillouin (WKB) approximation. There are two main challenges in solving the electron-plasmon kinetic system numerically. Firstly, non-uniform plasma density and magnetic field results in high dimensionality and the presence of multiple time scales. Secondly, a physically reliable numerical solution requires a structure-preserving scheme that enforces the conservation of mass, momentum, and energy. In this paper, we propose a struture-preserving multiscale solver for particle-wave interaction in non-uniform magnetized plasmas. The solver combines a conservative local discontinuous Galerkin (LDG) scheme for the interaction part with a trajectory averaging method for the plasmon Hamiltonian flow part. Numerical examples for a non-uniform magnetized plasma in an infinitely long symmetric cylinder are presented. It is verified that the LDG scheme rigorously preserves all the conservation laws, and the trajectory averaging method significantly reduces the computational cost.

A structure-preserving multiscale solver for particle-wave interaction in non-uniform magnetized plasmas

TL;DR

This work introduces a structure-preserving multiscale solver for particle–plasmon interactions in non-uniform magnetized plasmas, combining a conservative LDG discretization for diffusion–interaction terms with a trajectory-averaging approach for the fast plasmon Hamiltonian flow. It leverages trajectory bundles and a connection-proportion algorithm to discretize the kernel of the advection operator, ensuring exact mass, momentum, and energy conservation in the averaged system. The methodology yields substantial computational savings while maintaining fidelity to the underlying physics, as demonstrated by numerical tests in a non-uniform cylinder geometry that show position-dependent diffusion and strict conservation. The framework lays groundwork for large-scale runaway electron simulations and can be extended to more complex quasilinear models and geometries.

Abstract

Particle-wave interaction is of fundamental interest in plasma physics, especially in the study of runaway electrons in magnetic confinement fusion. Analogous to the concept of photons and phonons, wave packets in plasma can also be treated as quasi-particles, called plasmons. To model the ``mixture" of electrons and plasmons in plasma, a set of ``collisional" kinetic equations has been derived, based on weak turbulence limit and the Wentzel-Kramers-Brillouin (WKB) approximation. There are two main challenges in solving the electron-plasmon kinetic system numerically. Firstly, non-uniform plasma density and magnetic field results in high dimensionality and the presence of multiple time scales. Secondly, a physically reliable numerical solution requires a structure-preserving scheme that enforces the conservation of mass, momentum, and energy. In this paper, we propose a struture-preserving multiscale solver for particle-wave interaction in non-uniform magnetized plasmas. The solver combines a conservative local discontinuous Galerkin (LDG) scheme for the interaction part with a trajectory averaging method for the plasmon Hamiltonian flow part. Numerical examples for a non-uniform magnetized plasma in an infinitely long symmetric cylinder are presented. It is verified that the LDG scheme rigorously preserves all the conservation laws, and the trajectory averaging method significantly reduces the computational cost.

Paper Structure

This paper contains 24 sections, 11 theorems, 102 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The kinetic system for particles and plasmons
  3. Trajectorial average
  4. The Liouville-reaction equation
  5. The averaged kinetic system
  6. Trajectory bundle: definition and construction
  7. Trajectory bundles and their properties
  8. The connection-proportion algorithm
  9. The conservative scheme based on LDG method
  10. The bilinear form of local DG method
  11. Space discretization
  12. The conservative scheme
  13. Implementation and complexity
  14. Interpolated Hamiltonian
  15. Cut-off domains
  16. ...and 9 more sections

Key Result

Theorem 1

If $f(\mathbf{p}, \mathbf{x}, t)$ and $N(\mathbf{k}, \mathbf{x}, t)$ solve Equation(aversys) with emission/absorption kernel being replaced by $\mathcal{B}_{\varepsilon}$, then for any $\mathcal{B}_{\varepsilon}$ we have the following conservation laws,

Figures (4)

  • Figure 1: Trajectory bundles and their minimal triangle covers. The $x$-axis represents $r/R_{max}$, and the $y$-axis represents $k_{r}c/\omega_{0}$. Since the Hamiltonian is symmetric for $\pm k_{r}$, we only plot half of the domain. Note that $S_{11}$ and $S_{12}$ are two trajectory bundles generated by the same Hamiltonian range interval. And $S_{13}$ is not a single strip because it contains a saddle point.
  • Figure 2: Spatial inhomogeneity
  • Figure 3: Temporal evolution.
  • Figure 4: Conservation laws

Theorems & Definitions (24)

  • Theorem 1: unconditional conservation
  • proof
  • Definition 1
  • Definition 2: minimal triangle cover
  • Proposition 1
  • Corollary 1
  • Definition 3: connection matrix
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 14 more