An implicit discontinuous Galerkin finite element discrete Boltzmann method for high Knudsen number flows

Abstract

Simulations of the discrete Boltzmann Bhatnagar–Gross–Krook equation are an important tool for understanding fluid dynamics in non-continuum regimes. Here, we introduce a discontinuous Galerkin finite element method for spatial discretization of the discrete Boltzmann equation for isothermal flows with high 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 the 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.