Linear, Second order and Unconditionally Energy stable schemes for The Viscous Cahn-Hilliard Equation with hyperbolic relaxation using the invariant energy quadratization method

摘要

In this paper, we consider numerical approximations for the viscousCahn-Hilliard equation with hyperbolic relaxation. This type of equationsprocesses energy-dissipative structure. The main challenge in solving such adiffusive system numerically is how to develop high order temporaldiscretization for the hyperbolic and nonlinear terms, allowing largetime-marching step, while preserving the energy stability, i.e. the energydissipative structure at the time-discrete level. We resolve this issue bydeveloping two second-order time-marching schemes using the recently developed"Invariant Energy Quadratization" approach where all nonlinear terms arediscretized semi-explicitly. In each time step, one only needs to solve asymmetric positive definite (SPD) linear system. All the proposed schemes arerigorously proven to be unconditionally energy stable, and the second-orderconvergence in time has been verified by time step refinement testsnumerically. Various 2D and 3D numerical simulations are presented todemonstrate the stability, accuracy and efficiency of the proposed schemes.