Discontinuous Galerkin Discretizations of the Boltzmann Equations in 2D: semi-analytic time stepping and absorbing boundary layers

TL;DR

This paper develops an efficient discontinuous Galerkin method with semi-analytic time stepping and absorbing boundary layers for simulating nearly incompressible flows via Boltzmann equations, improving stability and accuracy.

Contribution

It introduces a novel PML formulation and semi-analytic time discretization techniques for Boltzmann equations, enhancing computational efficiency in stiff regimes.

Findings

Effective absorption of nonlinear fluctuations with PML

Improved time step restrictions with semi-analytic methods

Validated accuracy and efficiency on flow test cases

Abstract

We present an efficient nodal discontinuous Galerkin method for approximating nearly incompressible flows using the Boltzmann equations. The equations are discretized with Hermite polynomials in velocity space yielding a first order conservation law. A stabilized unsplit perfectly matching layer (PML) formulation is introduced for the resulting nonlinear flow equations. The proposed PML equations exponentially absorb the difference between the nonlinear fluctuation and the prescribed mean flow. We introduce semi-analytic time discretization methods to improve the time step restrictions in small relaxation times. We also introduce a multirate semi-analytic Adams-Bashforth method which preserves efficiency in stiff regimes. Accuracy and performance of the method are tested using distinct cases including isothermal vortex, flow around square cylinder, and wall mounted square cylinder test cases.