Efficient, Accurate and Energy Stable Discontinuous Galerkin Methods for Phase Field Models of Two-Phase Incompressible Flows (Communications in Computational Physics, 2019)
The goal of this paper is to propose fully discrete local discontinuous Galerkin(LDG) finite element methods for the Cahn-Hilliard-Navier-Stokes (CHNS) equation,which are shown to be unconditionally energy stable. In details, using the convexsplitting principle, we first construct a first order scheme and a second order Crank-Nicolson scheme for time discretizations. The proposed schemes are shown to be unconditionally energy stable. Then, using the invariant energy quadratization (IEQ) approach, we develop a novel linear and decoupled first order scheme, which is easy toimplement and energy stable. In addition, a semi-implicit spectral deferred correction(SDC) method combining with the first order convex splitting scheme is employedto improve the temporal accuracy. Due to the local properties of the LDG methods,the resulting algebraic equations at the implicit level is easy to implement and can besolved in an explicit way when it is coupled with iterative methods. In particular, wepresent efficient and practical multigrid solvers to solve the resulting algebraic equations, which have nearly optimal complexity. Numerical experiments of the accuracyand long time simulations are presented to illustrate the high order accuracy in bothtime and space, the capability and efficiency of the proposed methods.
Keywords: Cahn-Hilliard-Navier-Stokes equation; local discontinuous Galerkin method; convex splitting; invariant energy quadratization; unconditional energy stability; spectral deferred correction method.