论文|Qiaolin He, Xiaoding Shi：Energy stable discontinuous Galerkin method for compressible Navier–Stokes–Allen–Cahn system

本文（Energy stable discontinuous Galerkin method for compressible Navier–Stokes–Allen–Cahn system）原载Commun Nonlinear Sci，由四川大学数学学院贺巧琳教授等科研人员创作，系四川大学“智慧法治”超前部署学科系列学术成果。后续会持续分享四川大学“智慧法治”超前部署学科系列学术成果，欢迎大家阅读。

Abstract: In this paper, we present a fully discrete local discontinuous Galerkin (LDG) finite ele- ment 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 ac- curacy. 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 accu- racy 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. 

Qiaolin He, Xiaoding Shi, Energy stable discontinuous Galerkin method for compressible Navier–Stokes–Allen–Cahn system，Commun Nonlinear Sci, 2021, 98, 105771. （论文下载）