Numerical Approximations of Phase Field Models Using a General Class of Linear Time-Integration Schemes (Communications in Computational Physics, 2021)
In this paper, we develop a new class of linear time-integration schemes forphase-field models. The newly proposed schemes extend the recently developed energy quadratization technique by introducing extra free parameters to further stabilizethe schemes and improve their accuracy. The freshly proposed schemes have severaladvantages. First of all, they are rather generic such that they apply to most existing phase-field models in the literature. The resulted schemes are also linear in time,which means only a linear system needs to be solved during each time marching step.Thus, it significantly reduces the computational cost. Besides, they are unconditionallyenergy stable such that a larger time step size is practical. What is more, the solutionexistence and uniqueness in each time step are guaranteed without any dependence onthe time step size. To demonstrate the generality of the proposed schemes, we applythem to several typical examples, including the widely-used molecular beam epitaxy(MBE) model, the Cahn-Hilliard equation, and the diblock copolymer model. Numerical tests reveal that the proposed schemes are accurate and efficient. This new familyof linear and unconditionally energy stable schemes provides insights in developingnumerical approximations for general phase field models.
Keywords: Phase field; linear scheme; energy stable; Cahn-Hilliard; diblock copolymer; molecular beam epitaxy growth.