Energy Stable Numerical Method for the TDGL Equation with the Reticular Free Energy in Hydrogel

Abstract

Here we focus on the numerical simulation of the phase separation about macromolecule microsphere composite (MMC) hydrogel. The model is based on time-dependent Ginzburg-Landau (TDGL) equation with the reticular free energy. An unconditionally energy stable difference scheme is proposed based on the convex splitting of the corresponding energy functional. In the numerical experiments, we observe that simulating the whole process of the phase separation requires a considerably long time. We also notice that the total free energy changes significantly in initial stage and varies slightly in the following time. Based on these properties, we apply the adaptive time stepping strategy to improve the computational efficiency. It is found that the application of time step adaptivity can not only resolve the dynamical changes of the solution accurately but also significantly save CPU time for the long time simulation.

An energy stable finite element scheme for the three-component Cahn-Hilliard-type model for macromolecular microsphere composite hydrogels

“…Subsequently, a convex splitting method was presented in [35], and energy stability was proven for the numerical solution of the phase variable. Liao et al applied an adaptive time step strategy to improve computational efficiency in [36] and Dong et al [18,19] presented the theoretical analysis for the first and second-order energy stable schemes. A stabilized method was also used to solve the binary system by Xu et al in [50], though a theoretical justification of the stabilizing parameters (for energy decay) has not been established.…”