Linear, Second order and Unconditionally Energy stable schemes for The Viscous Cahn-Hilliard Equation with hyperbolic relaxation using the invariant energy quadratization method
This work provides efficient and stable numerical schemes for simulating phase-field models with hyperbolic relaxation, enabling larger time steps while preserving energy dissipation.
The authors developed two second-order unconditionally energy-stable time-marching schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation using the invariant energy quadratization method, requiring only the solution of a symmetric positive definite linear system per time step. Numerical tests verified second-order convergence and stability.
In this paper, we consider numerical approximations for the viscous Cahn-Hilliard equation with hyperbolic relaxation. This type of equations processes energy-dissipative structure. The main challenge in solving such a diffusive system numerically is how to develop high order temporal discretization for the hyperbolic and nonlinear terms, allowing large time-marching step, while preserving the energy stability, i.e. the energy dissipative structure at the time-discrete level. We resolve this issue by developing two second-order time-marching schemes using the recently developed "Invariant Energy Quadratization" approach where all nonlinear terms are discretized semi-explicitly. In each time step, one only needs to solve a symmetric positive definite (SPD) linear system. All the proposed schemes are rigorously proven to be unconditionally energy stable, and the second-order convergence in time has been verified by time step refinement tests numerically. Various 2D and 3D numerical simulations are presented to demonstrate the stability, accuracy and efficiency of the proposed schemes.