Discontinuous Galerkin Discretizations of the Boltzmann Equations in 2D: semi-analytic time stepping and absorbing boundary layers
For computational fluid dynamics, this provides a more efficient and stable method for simulating nearly incompressible flows with the Boltzmann equations.
The paper presents a discontinuous Galerkin method for the Boltzmann equations in 2D, using Hermite polynomials and a PML formulation for absorbing boundary layers, with semi-analytic time stepping to handle stiffness. Tests on vortex and cylinder flows show accuracy and efficiency.
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.