A High-Order Oscillation-Free Discontinuous Galerkin Scheme for One-Dimensional Compressible Multi-Material Flows

Abstract

In this paper, a high-order oscillation-free (OF) discontinuous Galerkin (DG) scheme is presented for one-dimensional compressible multi-material flows. For describing the dynamics of fluid mixture, we couple a conservative equation related to the volume-fraction model with the Euler equations. For controlling the oscillations, some damping terms are added into the weak formulation of the system to automatically adjust the high-order terms. There are not any parameters which need to be adjusted artificially in the new damping terms, and the difficulties in solving discontinuous solutions and complexities of designing limiters can be avoided. Our scheme can be applied to the simulation of compressible multi-material flows efficiently with the essentially non-oscillatory property. Moreover, our scheme can be extended to the one with any high order as long as the order of basis functions is increased. In this paper, we only study the third-order OFDG scheme with the basis functions  up to the quadratic polynomial. Some examples are tested to demonstrate the third-order accuracy and essentially non-oscillatory property of our scheme.