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A structure-preserving local discontinuous Galerkin method for the Fokker-Planck-Landau equation

Kun Huang, Andrés Galindo-Olarte, Rodrigo González-Hernández, Irene M. Gamba

TL;DR

This work develops a structure-preserving local discontinuous Galerkin method for the non-local, nonlinear Fokker-Planck-Landau equation by reformulating the collision operator as a nonlocal advection–diffusion problem and introducing a discrete gradient within an LDG framework. It then designs symmetric and upwind LDG schemes that preserve mass, momentum, and energy at the discrete level while ensuring entropy dissipation, achieved in part through a carefully chosen discrete collision kernel $\Phi_h$ and projection of the energy functional $\mathcal{E}$. The authors also leverage a variational-crime-based approximation to enforce the operator’s geometric structure and, for the non-relativistic case, exploit a convolution structure to drastically reduce storage and computation. Numerical experiments validate conservation properties and entropy decay, demonstrate the method’s high-order accuracy and stability, and indicate its potential for scalable, parallel implementations and extensions to relativistic FPL and Vlasov coupling.

Abstract

In this work, we introduce a structure-preserving local discontinuous Galerkin (LDG) method \cite{cockburn1998local} for solving the non-local non-linear Fokker-Planck-Landau (FPL) equations. We rephrase the structure-preserving strategy of Shiroto and Sentoku\cite{shiroto2019structure} in the language of numerical analysis, and extend it to the LDG framework. We propose a method that is not only conservative, but also stabilized through upwind flux. The apparent contradiction between conservation laws and numerical stabilization is elegantly resolved by leveraging the properties of the jump terms inherent to the LDG framework. In the numerical experiments, our scheme is tested with benchmark examples.

A structure-preserving local discontinuous Galerkin method for the Fokker-Planck-Landau equation

TL;DR

This work develops a structure-preserving local discontinuous Galerkin method for the non-local, nonlinear Fokker-Planck-Landau equation by reformulating the collision operator as a nonlocal advection–diffusion problem and introducing a discrete gradient within an LDG framework. It then designs symmetric and upwind LDG schemes that preserve mass, momentum, and energy at the discrete level while ensuring entropy dissipation, achieved in part through a carefully chosen discrete collision kernel and projection of the energy functional . The authors also leverage a variational-crime-based approximation to enforce the operator’s geometric structure and, for the non-relativistic case, exploit a convolution structure to drastically reduce storage and computation. Numerical experiments validate conservation properties and entropy decay, demonstrate the method’s high-order accuracy and stability, and indicate its potential for scalable, parallel implementations and extensions to relativistic FPL and Vlasov coupling.

Abstract

In this work, we introduce a structure-preserving local discontinuous Galerkin (LDG) method \cite{cockburn1998local} for solving the non-local non-linear Fokker-Planck-Landau (FPL) equations. We rephrase the structure-preserving strategy of Shiroto and Sentoku\cite{shiroto2019structure} in the language of numerical analysis, and extend it to the LDG framework. We propose a method that is not only conservative, but also stabilized through upwind flux. The apparent contradiction between conservation laws and numerical stabilization is elegantly resolved by leveraging the properties of the jump terms inherent to the LDG framework. In the numerical experiments, our scheme is tested with benchmark examples.

Paper Structure

This paper contains 17 sections, 5 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Fokker-Planck-Landau Equations
  3. The Local Discontinuous Galerkin Discretization
  4. Notations, definitions and projections
  5. The bilinear form of the LDG method
  6. Structure-Preserving Schemes
  7. Variational Crime
  8. Symmetric Structure-Preserving Scheme
  9. Upwind Structure-Preserving Scheme
  10. Implementation
  11. Evaluation and Complexity
  12. Temporal Discretizations
  13. Numerical Experiments
  14. The convolution structure of non-relativistic FPL operator
  15. Numerical Results
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose that $f(\mathbf{p},t)$ is the solution to the FPL equation eqn:fpl_mal, then we have

Figures (2)

  • Figure 1: Cross section plot of $f(\mathbf{p},t)$ at $p_{z}=0$ in different time. The simulation was performed on a $8\times8\times8$ mesh with $\mathbb{Q}^{2}$ basis functions.
  • Figure 2: Conservation errors and the decay of entropy. The simulation was performed on a $8\times8\times8$ mesh with $\mathbb{Q}^{2}$ basis functions.

Theorems & Definitions (16)

  • Theorem 1: Conservation Laws
  • proof
  • Remark 1
  • Definition 1: discrete gradient operator
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Remark 2
  • Theorem 3: Symmetric Structure-Preserving Scheme
  • ...and 6 more