A Slope Limiting Procedure in Discontinuous Galerkin Finite Element Method for Gasdynamics Applications

In this paper we demonstrate the performance of a slope limiting procedure combined with a discontinuous Galerkin (DG) finite element solver for 2D compressible Euler equations. The slope limiter can be categorized into van Albada type and is differentiable. This slope limiter is modified from a similar limiter used in finite volume solvers to suit the needs of the DG solver. The gradient in an element is limited using the weighted average of the face gradients. The face gradients are obtained from the area-weighted average of the gradient on both sides of the faces. The slope limiting process is very suitable for meshes discretized by triangle elements. The HLLC (Harten, Lax and van Leer) or the local Lax-Friedrich (LLF) flux functions is used to compute the interface fluxes in the DG formulation. The second order TVD Runge-Kutta scheme is employed for the time integration. The numerical examples including transonic, supersonic and hypersonic flows show that the current slope limiting process together with the DG solver is able to remove overshoots and undershoots around high gradient regions while preserving the high accuracy of the DG method. The convergence histories of all examples demonstrate that the limiting process does not stall convergence to steady state as many other slope limiters do.