A Simple Method to Improve the Accuracy of Advection in Discontinuous Galerkin Methods for Navier-Stokes Simulations

摘要

Discontinuous Galerkin (DG) methods have seen increased use over the past few decades. Recent work has shown that extremely accurate solutions (3p + 2) can be achieved for pure diffusion problems when using Recovery Discontinuous Galerkin (RDG) methods. However, the order of accuracy for advection is at best 2p+1, thus negating the main advantages of RDG, particularly for advection-diffusion equations, such as the Navier-Stokes equations. In the present paper, we determine a new approach based on monotone upstream centered schemes for conservation laws (MUSCL) to improve the accuracy of advection. Furthermore, we show that RDG is more computationally efficient than certain commonly used diffusion methods, thus making RDG a more viable approach for high-fidelity turbulence simulations. For this purpose, we use three test problems. The interaction of two rarefactions is considered to evaluate convergence of the new advection approach. The problem of decaying isotropic turbulence with eddy shocklets is used to demonstrate the improvement of the proposed approach over standard advection. A pure diffusion Navier-Stokes problem is employed to compare the performance of RDG against other commonly used diffusion schemes.