LAUSR

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.