A Spectral-Lagrangian Boltzmann Solver for a Multi-Energy Level Gas

TL;DR

This paper introduces a spectral-Lagrangian method for solving the Boltzmann equation in multi-energy level gases, accurately modeling elastic and inelastic collisions with a general, Fourier-based computational approach.

Contribution

It develops a general spectral-Lagrangian algorithm for the Boltzmann equation that handles multiple energy levels and inelastic collisions, with no restrictions on cross-section models.

Findings

Accurate numerical solutions for space homogeneous and inhomogeneous problems.

Validation against DSMC method shows high accuracy.

Conservation laws are enforced through constrained optimization.

Abstract

In this paper a spectral-Lagrangian method for the Boltzmann equation for a multi-energy level gas is proposed. Internal energy levels are treated as separate species and inelastic collisions (leading to internal energy excitation and relaxation) are accounted for. The formulation developed can also be used for the case of a mixture of monatomic gases without internal energy (where only elastic collisions occur). The advantage of the spectral-Lagrangian method lies in the generality of the algorithm in use for the evaluation of the elastic and inelastic collision operators. The computational procedure is based on the Fourier transform of the partial elastic and inelastic collision operators and exploits the fact that these can be written as weighted convolutions in Fourier space with no restriction on the cross- section model. The conservation of mass, momentum and energy during collisions is enforced through the solution of constrained optimization problems. Numerical solutions are obtained for both space homogeneous and space in- homogeneous problems. Computational results are compared with those obtained by means of the DSMC method in order to assess the accuracy of the proposed spectral-Lagrangian method.