Weighted ghost fluid discontinuous Galerkin method for two-medium problems

Abstract

A new interface treating method is proposed to simulate compressible two-medium problems with the Runge-Kutta discontinuous Galerkin (RKDG) method. In the present work, both the Euler equation and the level-set equation are discretized with the RKDG method which is compact and of high-order accuracy. The linearized interface inside an interface cell is recovered by the level-set function. The new solution of this cell is taken as a convex combination of two auxiliary solutions. One is the solution obtained by the RKDG method for a single-medium cell with proper numerical fluxes, and the other one is the intermediate state of the two-medium Riemann problem constructed in the normal direction. The weights of the two auxiliary solutions are carefully chosen according to the location of the interface inside the cell. Thus, it ensures a smooth transition when the interface leaves one cell and enters a neighboring cell. The entropy-fix technique is adopted to minimize the overshoots or undershoots in problems with large entropy ratio across the interface. The scheme is justified in a 1-dimensional situation and extended to 2-dimensional problems. Several 1-dimensional two-medium problems, including both smooth and discontinuous examples, are simulated and compared with exact solutions. Also, three 2-dimensional benchmark problems are simulated to validate the present method in two-medium problems.