Three-dimensional discontinuous Galerkin based high-order gas-kinetic scheme and GPU implementation

In this paper, the discontinuous Galerkin based high-order gas-kinetic schemes (DG-HGKS) are developed for the three-dimensional Euler and Navier-Stokes equations. Different from the traditional discontinuous Galerkin (DG) methods with Riemann solvers, the current method adopts a kinetic evolution process, which is provided by the integral solution of Bhatnagar-Gross-Krook (BGK) model. In the weak formulation of DG method, a time-dependent evolution function is provided, and both inviscid and viscous fluxes can be calculated uniformly. The temporal accuracy is achieved by the two-stage fourth-order discretization, and the second-order gas-kinetic solver is adopted for the fluxes over the cell interface and the fluxes inside a cell. Numerical examples, including accuracy tests and Taylor-Green vortex problem, are presented to validate the efficiency and accuracy of DG-HGKS. Both optimal convergence and super-convergence are achieved by the current scheme. The comparison between DG-HGKS and high-order gas-kinetic scheme with weighted essential non-oscillatory reconstruction (WENO-HGKS) is also given, and the numerical performances are comparable with the approximate number of degree of freedom. To accelerate the computation, the DG-HGKS is implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). The obtained results are also compared with those calculated by the central processing units (CPU) code in terms of the computational efficiency. The speedup of GPU code suggests the potential of high-order gas-kinetic schemes for the large scale computation.