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Fully spectral scheme for the linear BGK equation on the whole space

Bastien Grosse

TL;DR

The paper develops a fully spectral, structure-preserving numerical scheme for the linear inhomogeneous BGK equation on the real line with a confinement potential $\phi$. It combines a velocity Hermite expansion with a space-domain orthonormal polynomial basis to yield a semi-discrete system that retains discrete conservation laws and hypocoercivity, with exponential decay guaranteed by carefully constructed entropy functionals. The analysis covers both harmonic and non-harmonic potentials, providing hypocoercivity constants that depend on the truncation but are independent of the live discretization in the velocity/space expansions. Numerical experiments on harmonic and double-well potentials demonstrate robust long-time behavior, preservation of invariants, and clear exponential relaxation of perturbations, validating the approach for unbounded domains and long-time kinetic simulations.

Abstract

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $φ$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-φ}$ in space. The potential $φ$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.

Fully spectral scheme for the linear BGK equation on the whole space

TL;DR

The paper develops a fully spectral, structure-preserving numerical scheme for the linear inhomogeneous BGK equation on the real line with a confinement potential . It combines a velocity Hermite expansion with a space-domain orthonormal polynomial basis to yield a semi-discrete system that retains discrete conservation laws and hypocoercivity, with exponential decay guaranteed by carefully constructed entropy functionals. The analysis covers both harmonic and non-harmonic potentials, providing hypocoercivity constants that depend on the truncation but are independent of the live discretization in the velocity/space expansions. Numerical experiments on harmonic and double-well potentials demonstrate robust long-time behavior, preservation of invariants, and clear exponential relaxation of perturbations, validating the approach for unbounded domains and long-time kinetic simulations.

Abstract

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential , even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight in space. The potential is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.

Paper Structure

This paper contains 35 sections, 21 theorems, 211 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Notations and assumptions
  4. Main results
  5. Brief review of litterature and main features on the present work.
  6. Outline of the paper
  7. Continuous setting
  8. Decomposition on Hermite polynomials in velocity
  9. Conservation laws
  10. Semi-discrete spectral scheme in space and velocity
  11. Projection on orthonormal polynomials in space
  12. Discrete conservation laws
  13. Totally discrete scheme
  14. Time discretization
  15. Discrete conservation laws
  16. ...and 20 more sections

Key Result

Theorem 1.1

There exists two positive constants $C,\kappa$ such that for any solution $f\in \mathcal{C}(\mathbb{R}^{+},L^{2}(\mathbb{R}\times\mathbb{R},\mathcal{M}(x,v)dxdv))$ of (equation/edp1) with initial condition $f_{0}\in L^{2}(\mathbb{R}\times\mathbb{R},\mathcal{M}(x,v)dxdv)$,

Figures (3)

  • Figure 1: Exponential decay of the norm $\|\tilde{h} \|_{L^{2}(\mathcal{M})}$ for different parameter $K$ and $N$ (y-axis in logscale).
  • Figure 2: Exponential decay of the norm $\|\tilde{h} \|_{L^{2}(\mathcal{M})}$ for different parameter $K$ and $N$ (y-axis in logscale).
  • Figure 3: Evolution of the perturbation in the $(x,v)$-plane.

Theorems & Definitions (40)

  • Theorem 1.1: macroscopicmode
  • Theorem 1.2
  • Proposition 2.1: macroscopicmode
  • proof
  • Remark 3.1
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • Lemma 5.1
  • ...and 30 more