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Discrete hypocoercive estimates for discontinuous Galerkin methods: application to the Vlasov-Poisson-Fokker-Planck system

Yi Cai, Alain Blaustein, Tao Xiong, Francis Filbet

Abstract

We develop and analyze a class of structure-preserving discontinuous Galerkin schemes for the nonlinear Vlasov-Poisson-Fokker-Planck model, reformulated as a hyperbolic system through a Hermite expansion in the velocity variable. We discretize the Vlasov-Fokker-Planck equation with the discontinuous Galerkin method, while the Poisson equation is approximated with either a discontinuous Galerkin method or a Raviart-Thomas mixed finite element method. We prove the exponential relaxation to equilibrium for suitable initial data, uniformly with respect to the discretization parameters thanks to discrete hypocoercivity arguments. Moreover, we check that the resulting semi-discrete schemes preserve the physical invariants along with the L 2 variational structure of the linearized model. Numerical simulations verify the accuracy and the long-time behavior of the scheme.

Discrete hypocoercive estimates for discontinuous Galerkin methods: application to the Vlasov-Poisson-Fokker-Planck system

Abstract

We develop and analyze a class of structure-preserving discontinuous Galerkin schemes for the nonlinear Vlasov-Poisson-Fokker-Planck model, reformulated as a hyperbolic system through a Hermite expansion in the velocity variable. We discretize the Vlasov-Fokker-Planck equation with the discontinuous Galerkin method, while the Poisson equation is approximated with either a discontinuous Galerkin method or a Raviart-Thomas mixed finite element method. We prove the exponential relaxation to equilibrium for suitable initial data, uniformly with respect to the discretization parameters thanks to discrete hypocoercivity arguments. Moreover, we check that the resulting semi-discrete schemes preserve the physical invariants along with the L 2 variational structure of the linearized model. Numerical simulations verify the accuracy and the long-time behavior of the scheme.

Paper Structure

This paper contains 17 sections, 9 theorems, 153 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Discretization of Vlasov-Fokker-Planck equation
  3. Hermite decomposition for the velocity variable
  4. Discontinuous Galerkin discretization for the spatial variable
  5. Discretization of Poisson equation
  6. Local discontinuous Galerkin method
  7. Raviart-Thomas method
  8. Trend to equilibrium and invariants for the discrete system
  9. Strong reformulation of the discrete system
  10. Main result and strategy
  11. Discrete properties
  12. Preliminary results
  13. Proof of Theorem \ref{['thm:vpfp1d_hermite_semidiscrete_main_theorem']}
  14. Numerical simulations
  15. Order of convergence
  16. ...and 2 more sections

Key Result

Theorem 4.1

Let $\tau_0, T_0>0$ be fixed and consider the solution $(D_h,E_h)$ to eq:vpfp1d_hermite_semidiscrete_riesz-eq:ops_bh_riesz. There exists a positive constant $\kappa$ such that, if the initial data satisfies then it holds for all $t\in \mathbb{R}^+$ The constant $\kappa$ depends only on the temperature $T_0$, the domain length $\left|\mathbb{T}\right|$, the polynomial degree $m$, and the mesh quas

Figures (2)

  • Figure 5.1: Strong Landau damping: snapshots of the distribution function $f$ at $t \in \{4,16,40\}$ across the collisional regimes. $\tau_0\in\{10^1,10^3,10^5\}$, $(N_x, N_H) = (128, 640)$.
  • Figure 5.2: Strong Landau damping: time evolution in logarithmic scale of (a) the potential energy, (b) the distance to kinetic equilibrium, (c) the distance to macroscopic equilibrium, and (d) the distance to local equilibrium across the collisional regimes. $\tau_0\in\{10^1,10^3,10^5\}$, $(N_x,N_H) = (128,640)$.

Theorems & Definitions (15)

  • Theorem 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • Proposition 4.4
  • Lemma 4.5
  • proof
  • Lemma 4.6
  • proof
  • Lemma 4.7
  • ...and 5 more