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arXiv · 2502.14396

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

Bastien Grosse
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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 $\Phi$, 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 $e^{-$\Phi$}$ in space. The potential $\Phi$ 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.

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Bastien Grosse. 2025-02-20. Fully spectral scheme for the linear BGK equation on the whole space. https://arxiv.org/abs/2502.14396

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