会议日程

2020年11月15日上午，地点：腾讯会议   203749593
Session   1  Chair: 杨霄锋
08:30—09:00	何晓明（Missouri   University of Science and Technology） A   fully decoupled iterative method with 3D anisotropic immersed finite elements   of non-homogeneous flux jump for Kaufman-type discharge problems
09:00—9:30	徐岩（中国科学技术大学） Positivity-preserving   well-balanced arbitrary Lagrangian-Eulerian  discontinuous Galerkin   methods for the shallow water equations
9:30—10:00	于海军（中国科学院计算数学与科学工程计算研究所） Efficient   Energy Stable Schemes for Time-Fractional Phase-Field Systems
10:00—10:20	休息
Session   2  Chair: 陈传军
10:20—10:50	杨霄锋（University   of South Carolina） Efficient   numerical schemes for gradient flow models
10:50—11:20	杨将（南方科技大学） Energy   Dissipation and Numerical Methods for the Time-Fractional Allen-Cahn   Equations
11:20—11:50	冯新龙（新疆大学） Binary   Fluid-Surfactant Phase Field Model Coupled with Geometric Curvature on the   Curved Surface
会议结束

 

报告题目和摘要

 

A fully decoupled iterative method with 3D anisotropic immersed finite elements of non-homogeneous flux jump for Kaufman-type discharge problems

Xiaoming He

Missouri University of Science and Technology

In order to simulate the Kaufman-type discharge problems, a fully decoupled iterative method with anisotropic immersed finite elements on Cartesian meshes is proposed, especially for a three-dimensional (3D) non-axisymmetric anisotropic hybrid model which is more difficult than the axisymmetric or isotropic models. The classical hybrid model, which describes the important plasma distribution of the Kaufman-type discharge problems, couples several difficult equations together to form a large scale system. The 3D non-axisymmetric and anisotropic properties will further increase the complexity of this system. Hence it generally needs to be solved in the decoupled way for significantly reducing the computational cost. Based on the Particle-in-Cell Monte Carlo collision (PIC-MCC) method and the immersed finite element (IFE) method, we propose a fully decoupled iterative method for solving this complex system. The IFE method allows Cartesian meshes for general interface problems, while the traditional finite element methods require body-fitting meshes which are often unstructured. Compared with traditional finite element methods, this feature significantly improves the efficiency of the proposed 3D fully decoupled iterative method, while maintaining the optimal accuracy of the chosen finite elements. Numerical simulations of traditional Kaufman ion thruster and annular ion thruster discharge chambers are provided and compared with the corresponding lab experiment results to illustrate the features of the proposed method.

 

 

Positivity-preserving well-balanced arbitrary Lagrangian-Eulerian  discontinuous Galerkin methods for the shallow water equations

徐岩

中国科学技术大学

In this paper, we develop well-balanced arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods for the shallow water equations, which preserve not only the still water equilibrium, but also the moving water equilibrium. Based on the time dependent linear affine mapping, the ALE-DG method for conservation laws maintains almost all mathematical properties of DG methods on static grids, such as conservation, geometric conservation law (GCL), entropy stability and optimal error estimates. The main difficulty to obtain the well-balanced property of the ALE-DG method for shallow water equations is that the grid movement and usual time discretization may destroy the equilibrium. By adopting the GCL preserving Runge-Kutta methods and the techniques of well-balanced DG schemes on static grids, we successfully construct the high order well-balanced ALE-DG schemes for the shallow water equations.  Numerical experiments in different circumstances are provided to illustrate the well-balanced property and accuracy of these schemes.

 

 

 

 

 

 

 

Efficient Energy Stable Schemes for Time-Fractional Phase-Field Systems

于海军

中国科学院计算数学与科学工程计算研究所

 

For the time-fractional phase field models, the corresponding energy dissipation law has not been settled on both the continuous level and the discretelevel. In this work, we shall address this open issue. More precisely, we prove that the time-fractional phase field models indeed admit an energy dissipation law of an integral type. In the discrete level, we propose a class of finite difference schemes that can inherit the theoretical energy stability.  Our discussion covers the time-fractional gradient systems including the time-fractional Allen-Cahn equation, the time-fractional Cahn-Hilliard equation and the time-fractional molecular beam epitaxy models.  Moreover, a numerical study of the coarsening rate of random initial states depending on the fractional parameter $\alpha$ reveals that there are several coarsening stages for both time-fractional Cahn-Hilliard equation and

time- fractional molecular beam epitaxy model, while there exists a $-\alpha/3$ power law coarsening stage.

 

 

Efficient numerical schemes for gradient flow models

杨霄锋

University of South Carolina

 

The main challenge of constructing energy-stable numerical schemes for the gradient flow type of models with high stiffness is how to design proper temporal discretizations for the nonlinear terms. We develop the novel Invariant Energy Quadratization (IEQ) and Scalar Auxiliary Variable (SAV) approaches where the nonlinear potentials are transformed into the quadratic form and then discretized semi-implicitly. In these ways, one only needs to solve a linear and symmetric positive definite system for the IEQ method and two linear equations with constant coefficients for the SAV method. We also discuss how to apply these algorithms to complicated models including the anisotropic dendritic solidification model with and without the melt convection.  Various 2D and 3D numerical simulations are performed to demonstrate the stability and accuracy of the developed algorithms thereafter.

 

Energy Dissipation and Numerical Methods for the Time-Fractional Allen-Cahn Equations

杨将

南方科技大学

 

We consider a time-fractional Allen-Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order $\alpha\in(0,1)$. First, the well-posedness and (limited) smoothing property are systematically analyzed. We also prove two kinds of energy dissipation properties, precisely, the weighted nonlocal energy decay and fractional-in-time energy law. Second, after discretizing the fractional derivative by backward Euler convolution quadrature, we develop several unconditionally solvable and stable time stepping schemes, i.e., convex splitting scheme, weighted convex splitting scheme and linear weighted stabilized scheme. Meanwhile, we study the discrete energy dissipation property (in a weighted average sense), which is important for gradient flow type models. Finally, by using a discrete version of fractional Gr\"onwall's inequality and maximal $\ell^p$ regularity, we prove that the convergence rates of those time-stepping schemes are $O(\tau^\alpha)$ without any extra regularity assumption on the solution. We also present extensive numerical results to support our theoretical findings and to offer new insight on the time-fractional Allen-Cahn dynamics.

 

 

 

 

 

Binary Fluid-Surfactant Phase Field Model Coupled with Geometric Curvature on the Curved Surface

冯新龙

新疆大学

 

In this report, we introduce the numerical simulation of the binary fluid-surfactant phase field model on the curved surface. First, considering the effect of curvature on the system, the coupling term of curvature and binary fluid system is added to the free energy of the system, and a new phase field model is constructed. Based on the scalar auxiliary variable method, we study the numerical schemes, which consist of a surface finite element method for the spatial discretization, and first- and second-order implicit schemes for the temporal discretization. The unconditional energy stability of the first-order one is also strictly proved.Furthermore, we studied the binary fluid-surfactant phase field model with fluid flow on the curved surface, and also explored the influence of curvature on the system. The theoretical predictions are verified through numerical examples, and important characteristics of these two systems are observed: one is that the curvature can cause one of the two fluids to favor or avoid areas of high curvature on the curved surface;the other one is that the higher the coupling intensity is, the faster the energy reaches stability.