Stable boundary conditions for the Hermite Discretization of Boltzmann Equation in Multi Physical Space Dimensions

TL;DR

This paper develops stable boundary conditions for Hermite discretization of the multidimensional Boltzmann equation, ensuring accurate numerical solutions in complex physical and velocity spaces.

Contribution

It extends boundary condition techniques to multi-dimensional Boltzmann equations, introduces an algorithm for symmetrizer construction, and proposes Onsager boundary conditions for higher-order Hermite discretizations.

Findings

Stable boundary conditions ensure numerical stability.

The proposed Onsager boundary conditions improve solution accuracy.

Comparison shows better performance than Maxwell's model.

Abstract

Any numerical method fails to provide us with acceptable results if not equipped with appropriate boundary conditions. Catering to more realistic applications, in the present article we have extended the work done on the one plus one dimensional Boltzmann equation to the Boltzmann equation involving multi-dimensions in physical and velocity space. Criteria for stable boundary conditions, using energy estimates, have been discussed for linear symmetric hyperbolic initial boundary value problems. Since the use of energy estimates requires the hyperbolic system to be symmetric, the symmetric hyperbolicity of the moment equations arising from a Hermite discretization of the Boltzmann equation has been studied. Furthermore, an algorithm to construct a general symmetrizer for an arbitrary order Hermite discretization has been presented. A block structure for the multi-dimensional moment equations has been recognised which has been used to construct stable Onsager boundary conditions. The newly proposed Onsager boundary conditions have been used to study a Poisson heat conduction problem using a higher order Hermite discretization; the results have been compared to those obtained from the Maxwell's accommodation model.