Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

Abstract

Abstract. In order to suppress the failure of preserving positivity of density or pressure, a positivity-preserving limiter technique coupled with h-adaptive Runge-Kutta discontinuous Galerkin (RKDG) method is developed in this paper. Such a method is implemented to simulate flows with the large Mach number, strong shock/obstacle interactions and shock diffractions. The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented. This approach directly uses the cell solution polynomial of DG finite element space as the interpolation formula. The method is validated by the well documented test examples involving unsteady compressible flows through complex bodies over a large Mach numbers. The numerical results demonstrate the robustness and the versatility of the proposed approach.

A high‐order Runge‐Kutta discontinuous Galerkin method with a subcell limiter on adaptive unstructured grids for two‐dimensional compressible inviscid flows

“…It turns out that, to simulate this problem, only about 3.76% of effective grid resolution needs to be used by the AMR strategy. The result matches well with the same reported in the work of Liu et al…”

“…We simulate a Mach 10 shock wave diffraction at a convex corner, which is a well‐studied benchmark problem in computational fluid dynamics. It is challenging to develop a stable numerical scheme for a high‐order RKDG due to the development of a low pressure region near to the 120 o convex corner . We consider a right‐moving Mach 10 shock initially located at x = 3.4, moving in an undisturbed fluid medium with ρ =1.4 and p =1.…”

“…The purpose of this test case is to demonstrate the potential of the present scheme with adaptively refined unstructured grid for solving problems on complex domains. In this problem, we simulate a Mach 1.3 shock passing an equilateral triangular solid body . The domain of interest is the region [‐0.65,0.5]×[‐0.5,0.5] with a triangular body having the coordinates of its vertices (‐0.2,0), (0.1,‐1/6) and (0.1,1/6).…”

Numerical simulation of compressible multifluid flows using an adaptive positivity‐preserving RKDG‐GFM approach

“…To remedy this situation, several positivity‐preserving limiters are constructed by Zhang et al For the present multifluid algorithm, as the material interface is invisible in all calculations by defining ghost fluid in the GFMs, the same positivity‐preserving limiter can be directly used as in single‐fluid flow. The modified positivity‐preserving limiter developed in the work of Liu et al and used under a CFL‐like condition is adopted in this work. For any cell K , the DG polynomials U K ( x ) and its average are To enforce the positivity of the density and pressure fields, a linear scaling limiter introduced by Liu and Osher is adopted.…”

Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations

“…In the CG version: calculate the mass-weighted averages using (13 by (40) and (37). Nonnegativity of the pressure p(U i ) corresponding to the final nodal values U i = (ρ i , (ρv) i , (ρE) i ) T of a continuous finite element approximation U h can be shown as in [18]. Since the pressure p(U ) is a concave function of the conserved variables, Jensen's inequality yields …”