Structural Characters and Isolated Stability of Phosphorus Polyanions

The optimized geometries at the RHF/6-311++G** level, the relatively stable energy at the MPW1PW91/6-311++G** level and the structural characters of anions have been acquired, indicating the stability is related to the chemical bonding of μ2?P atoms and the distri- bution of negative charges. The configurations of cage units P8 4- and P9 5- are stable due to the less torsion, but their ES values are relatively higher than that of P7 3- with more μ2?P atoms and the isolated stability is lower than that of P7 . They potentially play an important role as intermediate 3- in chemical reaction of producing complicated polyphosphides. Based on the related electronic properties, a stable polyanion must have low valence electron concentration, no (μ2?P)?(μ2?P) bond and a little dispersive charge. The earmark IR frequencies of cage units have been assigned to the vibration models in the end.

DECOUPLED, ENERGY STABLE SCHEME FOR HYDRODYNAMIC ALLEN-CAHN PHASE FIELD MOVING CONTACT LINE MODEL

In this paper, we present an efficient energy stable scheme to solve a phase field model incorporating contact line condition. Instead of the usually used Cahn-Hilliard type phase equation, we adopt the Allen-Cahn type phase field model with the static contact line boundary condition that coupled with incompressible Navier-Stokes equations with Navier boundary condition. The projection method is used to deal with the Navier-Stokes equa- tions and an auxiliary function is introduced for the non-convex Ginzburg-Landau bulk potential. We show that the scheme is linear, decoupled and energy stable. Moreover, we prove that fully discrete scheme is also energy stable. An efficient finite element spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes. Numerical results show that the proposed scheme is very efficient and accurate.

ENERGY STABLE NUMERICAL METHOD FOR THE TDGL EQUATION WITH THE RETICULAR FREE ENERGY IN HYDROGEL

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 Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon.The numerical simulation of the Cahn-Hilliardmodel needs very long time to reach the steady state,and therefore large time-stepping methods become useful.The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations.The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time.The proposed scheme is proved to be unconditionally energy stable and mass-conservative.An error estimate for the numerical solution is also obtained with second order in both space and time.By using this energy stable scheme,an adaptive time-stepping strategy is proposed,which selects time steps adaptively based on the variation of the free energy against time.The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

A High Order Adaptive Time-Stepping Strategy and Local Discontinuous Galerkin Method for the Modified Phase Field Crystal Equation

In this paper,we will develop a first order and a second order convex splitting,and a first order linear energy stable fully discrete local discontinuous Galerkin(LDG)methods for the modified phase field crystal(MPFC)equation.In which,the first order linear scheme is based on the invariant energy quadratization approach.The MPFC equation is a damped wave equation,and to preserve an energy stability,it is necessary to introduce a pseudo energy,which all increase the difficulty of constructing numerical methods comparing with the phase field crystal(PFC)equation.Due to the severe time step restriction of explicit timemarchingmethods,we introduce the first order and second order semi-implicit schemes,which are proved to be unconditionally energy stable.In order to improve the temporal accuracy,the semi-implicit spectral deferred correction(SDC)method combining with the first order convex splitting scheme is employed.Numerical simulations of the MPFC equation always need long time to reach steady state,and then adaptive time-stepping method is necessary and of paramount importance.The schemes at the implicit time level are linear or nonlinear and we solve them by multigrid solver.Numerical experiments of the accuracy and long time simulations are presented demonstrating the capability and efficiency of the proposed methods,and the effectiveness of the adaptive time-stepping strategy.

On a Large Time-SteppingMethod for the Swift-Hohenberg Equation

The main purpose of this work is to contrast and analyze a large timestepping numerical method for the Swift-Hohenberg(SH)equation.This model requires very large time simulation to reach steady state,so developing a large time step algorithm becomes necessary to improve the computational efficiency.In this paper,a semi-implicit Euler schemes in time is adopted.An extra artificial term is added to the discretized system in order to preserve the energy stability unconditionally.The stability property is proved rigorously based on an energy approach.Numerical experiments are used to demonstrate the effectiveness of the large time-stepping approaches by comparing with the classical scheme.

Efficient numerical algorithms for the phase field crystal equation

In this paper,based on the Lagrange Multiplier approach in time and the Fourierspectral scheme for space,we propose efficient numerical algorithms to solve the phase field crystal equation.The numerical schemes are unconditionally energy stable based on the original energy and do not need the lower bound hypothesis of the nonlinear free energy potential.The unconditional energy stability of the three semi-discrete schemes is proven.Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of our proposed schemes.