A finite element discrete Boltzmann method for high Knudsen number flows

TL;DR

This paper develops a high-order discontinuous Galerkin finite element method for simulating high Knudsen number flows using the discrete Boltzmann equation, validated through accuracy tests and flow simulations.

Contribution

It introduces a novel DG-FEM approach combined with high-order Runge-Kutta schemes for discrete Boltzmann equations at high Kn, enabling accurate and efficient simulations.

Findings

Validated high-order accuracy on Couette flow at Kn=1.

Compared Gauss-Hermite and Newton-Cotes quadratures for flow capture.

Demonstrated effective simulation of micro-cavity flows at high Kn.

Abstract

Simulations of the discrete Boltzmann Bhatnagar-Gross-Krook (BGK) equation are an important tool for understanding fluid dynamics in non-continuum regimes. Here, we introduce a discontinuous Galerkin finite element method (DG-FEM) for spatial discretization of the discrete Boltzmann equation for isothermal flows with Knudsen numbers (Kn~O(1)). In conjunction with a high-order Runge-Kutta time marching scheme, this method is capable of achieving high-order accuracy in both space and time, while maintaining a compact stencil. We validate the spatial order of accuracy of the scheme on a two-dimensional Couette flow with Kn = 1 and the D2Q16 velocity discretization. We then apply the scheme to lid-driven micro-cavity flow at Kn = 1, 2, and 8, and we compare the ability of Gauss-Hermite (GH) and Newton-Cotes (NC) velocity sets to capture the high non-linearity of the flow-field. While GH quadrature provides higher integration strength with fewer points, the NC quadrature has more uniformly distributed nodes with weights greater than machine-zero, helping to avoid the so-called ray-effect. Broadly speaking, we anticipate that the insights from this work will help facilitate the efficient implementation and application of high-order numerical methods for complex high Knudsen number flows.