Energy stable discontinuous Galerkin method for compressible Navier–Stokes–Allen–Cahn system

Abstract

In this paper, we present a fully discrete local discontinuous Galerkin (LDG) finite element method combined with scalar auxiliary variable (SAV) approach for the compressible Navier–Stokes–Allen–Cahn (NSAC) system. We start with a linear and first order scheme for time discretization and the minimal dissipation LDG for spatial discretization, which is based on the SAV approach and is proved to be unconditionally energy stable for one dimensional case. The velocity, the density and the mass concentration of fluid mixture can be solved separately. In addition, a semi-implicit spectral deferred correction (SDC) method combined with the first order scheme is employed to improve the temporal accuracy. Due to the local properties of the LDG methods, the resulting algebraic equations at the implicit level are easy to implement. In particular, we use efficient and practical multigrid solvers to solve the resulting algebraic equations. Although there is no proof of stability for the semi-implicit SDC with LDG spatial discretization, numerical experiments of the accuracy and long time simulations are presented to illustrate the high order accuracy in both time and space, the discretized energy stablity, the capability and efficiency of the proposed method. Numerical results show that the initial state determines the long time behavior of the diffusive interface for the two–phase flow, which are consistent with theoretical asymptotic stability results in Chen et al. (2018)[1].