Deterministic Solution of the Boltzmann Equation Using Discontinuous Galerkin Discretizations in Velocity Space

TL;DR

This paper introduces a deterministic discontinuous Galerkin method in velocity space for solving the Boltzmann equation, achieving high accuracy and efficiency, and validated through numerical comparisons with DSMC.

Contribution

It develops a novel DG-based deterministic approach with pre-computed kernels for the Boltzmann equation, applicable to any molecular potential.

Findings

Accurate solutions for spatially homogeneous relaxation of hard spheres.

Method shows excellent agreement with DSMC simulations.

Computational complexity scales as O(n^5) per phase space point.

Abstract

We present a new deterministic approach for the solution of the Boltzmann kinetic equation based on nodal discontinuous Galerkin (DG) discretizations in velocity space. In the new approach the collision operator has the form of a bilinear operator with pre-computed kernel; its evaluation requires $O(n^5)$ operations at every point of the phase space where $n$ is the number of degrees of freedom in one velocity dimension. The method is generalized to any molecular potential. Results of numerical simulations are presented for the problem of spatially homogeneous relaxation for the hard spheres potential. Comparison with the method of Direct Simulation Monte Carlo (DSMC) showed excellent agreement.