P-Multigrid Solution of Discontinuous Galerkin Discretizations of Euler Equations on Unstructured Meshs

The purpose of this paper is to simulate the transonic flow using the discontinuous Galerkin method associating with p-multigrid scheme. Usually, explicit temporal discretization such as multi-stage TVD Runge-Kutta schemes (TVD-RKDG) is used to advance the solution in time. However, for large-scale simulations and especially for high-order solutions, the rate of convergence slows down dramatically which is strictly restricted by CFL number, resulting in inefficient solution techniques to steady state solutions. To speed up convergence, a fast, low storage p-multigrid method is introduced in this article. Unlike the traditional p-multigrid methods where the same time integration scheme is used on all approximation levels, we use an explicit multi-stage Runge-Kutta scheme as the iterative smoother on the higher level approximations and a matrix-free implicit LU-SGS implicit method as the iterative smoother on the lowest level approximation. Numerical simulation for both 2D and 3D Euler Equations are presented to demonstrate the efficiency of the p-multigrid method. The results show that p-multigrid method could accelerate the convergence speed nearly one order of magnitude and maintain the original accuracy, compared with explicit forth-stage TVD Runge-Kutta method.