A High-Order Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for the Boltzmann Equation in Nearly Incompressible Flows

Abstract

We propose the arbitrary Lagrangian-Eulerian (ALE) form of the Galerkin-Boltzmann formulation for the simulation of nearly incompressible flows with moving boundaries. The continuous Boltzmann equations are mapped to a reference state to compensate the mesh motion with an advection term. The resulting system is discretized in space using the discontinuous Galerkin method on unstructured meshes. A semi-analytic Runge-Kutta time discretization is used to overcome the stiffness introduced by the continuous Boltzmann equations. The well-known geometric conservation law is shown to be satisfied by the time and space discretizations and consistent update of geometric factors of the discretization. The implementation is on the GPU accelerated kernel library libParanumal and validated by a free stream preservation and moving Taylor-Green vortex test cases. Then, the capabilities are shown using a plunging symmetric airfoil in two-dimensions and moving carangiform fish in three-dimensions using perfectly matched layers.