A parallel Runge–Kutta discontinuous Galerkin solver for rarefied gas flows based on 2D Boltzmann kinetic equations

Abstract

The high-order Runge–Kutta Discontinuous Galerkin (RKDG) method is applied to solve the 2D Boltzmann kinetic equations. A conservative DG type discretization of the non-linear collision relaxation term is formulated for both the Bhatnagar–Gross–Krook and the ellipsoidal statistical kinetic models. Verification is carried out for a steady and an unsteady oscillatory 1D Couette flows, a 2D conduction problem as well as for a 2D long microchannel flow by comparison with the DSMC and analytical solutions. The computational performance of the RKDG method is compared with a widely used second-order finite volume method. The RKDG method has up to 3rd-order spatial accuracy and up to 4th-order time accuracy and is more efficient than the finite volume approach. The parallelization by domain decomposition in physical space is implemented and parallel performance is evaluated. It is shown that 2nd order RKDG is over 15 times faster than the 2nd-order FVM method for the Couette flow test case. The high-order RKDG method is especially attractive for solution of low-speed and unsteady rarefied flows.