Zhengru Zhang

Lixiu Dong, Wenqiang Feng, Cheng Wang et al.

In this paper we analyze and implement a second-order-in-time numerical scheme for the three-dimensional phase field crystal (PFC) equation. The numerical scheme was proposed in [46], with the unique solvability and unconditional energy stability established. However, its convergence analysis remains open. We present a detailed convergence analysis in this article, in which the maximum norm estimate of the numerical solution over grid points plays an essential role. Moreover, we outline the detailed multigrid method to solve the highly nonlinear numerical scheme over a cubic domain, and various three-dimensional numerical results are presented, including the numerical convergence test, complexity test of the multigrid solver and the polycrystal growth simulation.

Jun Han, Hui Zhang, Zhengru Zhang

In this paper, we focus on the numerical simulation of phase separation about macromolecule microsphere composite (MMC) hydrogel. The model equation is based on Time-Dependent Ginzburg-Landau (TDGL) equation with reticular free energy. We have put forward two $L^2$ stable schemes to simulate simplified TDGL equation. In numerical experiments, we observe that simulating the whole process of phase separation requires a considerably long time. We also notice that the total free energy changes significantly in initial time and varies slightly in the following time. Based on these properties, we introduce an adaptive strategy based on one of stable scheme mentioned. It is found that the introduction of the time adaptivity cannot only resolve the dynamical changes of the solution accurately but also can significantly save CPU time for the long time simulation.

Rui Wang, Yunfeng Xiong, Zhengru Zhang

The chemotaxis PDE system with singular sensitivity was originally proposed by Short et al. (Math. Mod. Meth. Appl. Sci., 2008) as the continuum limit of a biased random walk model to account for the formation of crime hotspots and environmental feedback successfully. Recently, this idea has also been applied to epidemiology to model the impact of human social behaviors on disease transmission. In order to characterize the phase transition, pattern formation and statistical properties in the long-term dynamics, a stable and accurate numerical scheme is urgently demanded, which still remains challenging due to the positivity constraint on the singular sensitivity and the absence of an energy functional. In particular, the loss of positivity may produce nonphysical states and even cause spurious blow-up. To address these numerical challenges, this paper constructs an efficient positivity-preserving, implicit-explicit scheme with second-order accuracy. A rigorous error estimation is provided with the Lagrange multiplier correction to deal with the singular sensitivity. The whole framework is extended to a multi-agent epidemic model with degenerate diffusion, in which both positivity and mass conservation are achieved. Numerical experiments are performed to validate the theoretical results and demonstrate the necessity of the correction strategy. Our simulations reveal rich dynamical behaviors, including the phase transition between aggregation-dominated and dissipative regimes, as well as the nucleation, spread, and dissipation of crime hotspots. For the epidemic model, the results further show that spatial clustering of population density may accelerate virus transmission and significantly amplify the infectious wave.

Yingying Wang, Xiao Li, Zhengru Zhang

It is well known that the exponential time differencing (ETD) method has been successfully applied to the classic Cahn-Hilliard equation with double well potential. However, this numerical method can not be extended to the Cahn-Hilliard equation with Flory-Huggins potential directly due to the fact that the the numerical solution may go beyond the physical interval which leads the non-physical solution. In this paper, we develop and analyze first- and second-order numerical schemes for the nonlocal Cahn-Hilliard equation with the classic Flory-Huggins energy potential. In more detail, the ETD method is firstly used to obtain the prediction solution, and then this prediction solution is corrected by the projection method to avoid non-physical solution. The proposed method is shown to preserve bound and mass conservation in discrete settings. In addition, error estimates for the numerical solution are rigorously obtained for both schemes. Extensive numerical tests and comparisons are conducted to demonstrate the performance of the proposed schemes.

Wenqiang Feng, Cheng Wang, Steven M. Wise et al.

In this paper, we study a novel second-order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for the higher oder in time temporal discretization is how to ensure an unconditional energy stability and an efficient numerical implementation. We propose a general framework for designing the higher order in time numerical scheme with unconditional energy stability by using the BDF method with constant coefficient stabilized terms. Based on the unconditional energy stability property, we derive an $L^\infty_h (0,T; H_{h}^2)$ stability for the numerical solution and provide an optimal the convergence analysis. To deal with the 4-Laplacian solver in an $L^{2}$ gradient flow at each time step, we apply an efficient preconditioned steepest descent algorithm and preconditioned nonlinear conjugate gradient algorithm to solve the corresponding nonlinear system. Various numerical simulations are present to demonstrate the stability and efficiency of the proposed schemes and slovers.