Steady-State Simulation of Euler Equations by the Discontinuous Galerkin Method with the Jump Filter

Abstract

In this study, we introduce a jump filter for discontinuous Galerkin (DG) methods on both rectangular and triangular meshes, specifically designed for steady-state solutions of the Euler equations. Traditional limiters have occasionally been observed to produce slight post-shock oscillations, but the integration of the jump filter effectively reduces these fluctuations. This improvement allows the residual to stabilize at a level indistinguishable from machine zero in steady-state simulations. The jump filter operates by applying distinct damping factors to the different moments of the solution, based on the jump information at cell interfaces. This strategy creates transition zones from smooth regions to shock waves, which is critical for achieving steady-state solutions with residuals approaching the threshold of machine zero. The jump filter on unstructured meshes using Dubiner polynomials associated with the Sturm-Liouville operator is also investigated. Furthermore, the DG method with the jump filter retains many of the key advantages of the classical DG approach, including compactness, conservation, stability, and high-order accuracy. Extensive numerical experiments, conducted on both rectangular and triangular meshes, confirm the effectiveness of the jump filter in not only suppressing numerical oscillations but also significantly driving the residual towards machine zero.