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A Stable Arbitrarily High Order Time-Stepping Method for Thermal Phase Change Problems

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Communications in Computational Physics 2023年第2期33卷 477-510页

作者： Weiwen Wang Chuanju Xu School of Mathematical Sciences and Fujian Provincial Key Laboratory of MathematicalModeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005P.R.China.

Thermal phase change problems are widespread in mathematics,nature,and science.They are particularly useful in simulating the phenomena of melting and solidification in materials science.In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase changemodel,which is the coupling of a heat transfer equation and a phase field equation.The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes.A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step,with an efficient scheme of sufficient accuracy to calculate the solution at the first step.It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps.Adaptive time step size strategies can be applied to further benefit from this unconditional stability.Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.

关键词： Thermal phase change problem gradient flows unconditional energy stability auxiliary variable Runge-Kutta methods phase field

A New Conservative Allen-Cahn Type Ohta-Kawaski Phase-Field Model for Diblock Copolymers and Its Numerical Approximations

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Advances in Applied Mathematics and Mechanics 2022年第1期14卷 101-124页

作者： Shuang Geng Tongmao Li Qiongwei Ye Xiaofeng Yang College of Management Shenzhen UniversityShenzhenGuangdong 518060China Business School Yunnan University of Finance and EconomicsKunmingYunnan 650221China Department of Mathematics University of South CarolinaColumbiaSC 29208USA

We develop a new conservative Allen-Cahn phase-field model for diblock copolymers in this paper by using the Allen-Cahn type gradient flow approach for the classical Ohta-Kawaski free energy.The change in volume fraction of two composing monomers is eliminated by using a nonlocal Lagrange multiplier.Based on the recently developed stabilized Scalar Auxiliary Variable method,we have further developed an effective numerical scheme to solve the model.The scheme is highly efficient and only two linear and decoupled equations are needed to solve at every time step.We then prove that the numerical method is unconditionally energy stable,the stability and accuracy of the new scheme are demonstrated by numerous numerical examples conducted.By qualitatively comparing the equilibrium solution obtained by the new model and the classic Cahn-Hilliard model,we illustrate the effectiveness of the new model.

关键词： Phase-field Diblock copolymer Allen-Cahn nonlocal second order unconditional energy stability

Convergence Analysis of the Fully Decoupled Linear Scheme for Magnetohydrodynamic Equations

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Journal of Applied Mathematics and Physics 2022年第11期10卷 3462-3474页

作者： Zeyu Xia Qian Xu School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu China

In this paper, we propose a fully decoupled and linear scheme for the magnetohydrodynamic (MHD) equation with the backward differential formulation (BDF) and finite element method (FEM). To solve the system, we adopt a technique based on the “zero-energy-contribution” contribution, which separates the magnetic and fluid fields from the coupled system. Additionally, making use of the pressure projection methods, the pressure variable appears explicitly in the velocity field equation, and would be computed in the form of a Poisson equation. Therefore, the total system is divided into several smaller sub-systems that could be simulated at a significantly low cost. We prove the unconditional energy stability, unique solvability and optimal error estimates for the proposed scheme, and present numerical results to verify the accuracy, efficiency and stability of the scheme.

关键词： MHD Equations Zero-energy-Contribution Unique Solvability unconditional energy stability Optimal Error Estimates

Convergence Analysis of the Fully Decoupled Linear Scheme for Magnetohydrodynamic Equations

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Convergence Analysis of the Fully Decoupled Linear Scheme fo...