Volume 26, Issue 4
Efficient, Accurate and Energy Stable Discontinuous Galerkin Methods for Phase Field Models of Two-Phase Incompressible Flows

Ruihan Guo & Yan Xu

Commun. Comput. Phys., 26 (2019), pp. 1224-1248.

Published online: 2019-07

Preview Purchase PDF 32 5170
Export citation
  • Abstract

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 convex splitting 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 to implement and energy stable. In addition, a semi-implicit spectral deferred correction (SDC) method combining with the first order convex splitting 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 is easy to implement and can be solved in an explicit way when it is coupled with iterative methods. In particular, we present efficient and practical multigrid solvers to solve the resulting algebraic equations, which have nearly optimal complexity. Numerical experiments of the accuracy and long time simulations are presented to illustrate the high order accuracy in both time 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.

  • AMS Subject Headings

65M60, 35Q35, 35Q30, 35G25

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

rguo@zzu.edu.cn (Ruihan Guo)

yxu@ustc.edu.cn ( Yan Xu)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-26-1224, author = {Guo , Ruihan and Xu , Yan }, title = {Efficient, Accurate and Energy Stable Discontinuous Galerkin Methods for Phase Field Models of Two-Phase Incompressible Flows}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {4}, pages = {1224--1248}, abstract = {

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 convex splitting 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 to implement and energy stable. In addition, a semi-implicit spectral deferred correction (SDC) method combining with the first order convex splitting 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 is easy to implement and can be solved in an explicit way when it is coupled with iterative methods. In particular, we present efficient and practical multigrid solvers to solve the resulting algebraic equations, which have nearly optimal complexity. Numerical experiments of the accuracy and long time simulations are presented to illustrate the high order accuracy in both time and space, the capability and efficiency of the proposed methods.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0100}, url = {http://global-sci.org/intro/article_detail/cicp/13235.html} }
TY - JOUR T1 - Efficient, Accurate and Energy Stable Discontinuous Galerkin Methods for Phase Field Models of Two-Phase Incompressible Flows AU - Guo , Ruihan AU - Xu , Yan JO - Communications in Computational Physics VL - 4 SP - 1224 EP - 1248 PY - 2019 DA - 2019/07 SN - 26 DO - http://doi.org/10.4208/cicp.OA-2018-0100 UR - https://global-sci.org/intro/article_detail/cicp/13235.html KW - Cahn-Hilliard-Navier-Stokes equation, local discontinuous Galerkin method, convex splitting, invariant energy quadratization, unconditional energy stability, spectral deferred correction method. AB -

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 convex splitting 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 to implement and energy stable. In addition, a semi-implicit spectral deferred correction (SDC) method combining with the first order convex splitting 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 is easy to implement and can be solved in an explicit way when it is coupled with iterative methods. In particular, we present efficient and practical multigrid solvers to solve the resulting algebraic equations, which have nearly optimal complexity. Numerical experiments of the accuracy and long time simulations are presented to illustrate the high order accuracy in both time and space, the capability and efficiency of the proposed methods.

Ruihan Guo & Yan Xu. (2019). Efficient, Accurate and Energy Stable Discontinuous Galerkin Methods for Phase Field Models of Two-Phase Incompressible Flows. Communications in Computational Physics. 26 (4). 1224-1248. doi:10.4208/cicp.OA-2018-0100
Copy to clipboard
The citation has been copied to your clipboard