相场研究的最新进展研讨会
2020年11月14日, 北京
Zoom ID: 68490979428 Password: 445833
(Morning session: 美国东部时间11月13日7:00pm—11:00pm)
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8:00am-8:05am
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Opening
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8:05 am-8:55am
(7:05pm-7:55pm, Nov.13, US EST)
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Shen Jie (Purdue)
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8:55 am- 9:45 am
(7:55pm-8:45pm, Nov.13, US EST)
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Wang Xiaoming (SUSTEC)
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Break (15 min)
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10:00am-10:50am
(9:00 pm-9:50pm, Nov.13, US EST)
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Ju Lili (USC)
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10:50am-11:40am
(9:50pm-10:40pm, Nov.13, US EST)
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Yang Xiaofeng (USC)
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Lunch (11:40am-2:00pm)
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2:00pm-2:50pm
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Xu Chuanju (Xiamen U)
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2:50pm-3:40pm
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Xu Yan (USTC)
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Break (15 min)
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3:55pm-4:45pm
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Yu Haijun (CAS)
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4:45pm-5:35pm
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Li Xiao (PolyU)
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5:35pm-5:40pm
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Closing remark
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本次会议由北京师范大学2020数学科学学院双一流经费 团队项目“基于能量泛函的非线性梯度流模型的数值方法研究”资助
项目组成员：蔡勇勇，曹外香，陈华杰，崔丽，纪光华，刘君，潘亮，王成，张争茹
本次会议组织者：蔡勇勇，纪光华，王成，张争茹
Localized exponential time differencing methods: algorithms and analysis
Lili JU
University of South Carolina
Abstract: Exponential time differencing (ETD) has been proven to be very effective for solving stiff evolution problems in the past decades due to rapid development of matrix exponential algorithms and computing capacities. While direct parallelization of the ETD method is rarely of good efficiency due to the required data communication, the localized exponential time differencing approach was recently introduced for extreme-scale phase field simulations of coarsening dynamics, which displays excellent parallel scalability in modern supercomputers. The main idea is to use domain decomposition techniques to reduce the size of the problem, so that one instead only solves a group of smaller-sized subdomain problems simultaneously using the locally computed products of matrix exponentials and vectors. With the diffusion equation as the model problem, we will develop and analyze some overlapping and nonoverlapping localized ETD schemes and their solution algorithms. Numerical experiments are also carried out to confirm the theoretical results. In addition, extension of the LETD method to semilinear parabolic equations will be briefly discussed.
Maximum bound principles for a class of semilinear parabolic equations and their exponential time differencing schemes
Xiao LI
The Hongkong Polytechnic University
Abstract： A large class of semilinear parabolic equations share a practically desirable property, that is, the maximum bound principle (MBP) in the sense that if the absolute values of initial and boundary conditions are bounded by some constant, then the absolute value of the solution is also bounded pointwise by the same constant for all time. In this talk, we begin with the 1-D Allen-Cahn equation and its MBP-preserving exponential time differencing (ETD) schemes. Then we generalize these results to establish an abstract framework on general semilinear parabolic equations and give sufficient conditions on the linear and nonlinear operators that lead to such an MBP. We conclude that the first- and second-order stabilized ETD schemes preserve the MBPs unconditionally. The framework is then applied to nonlocal Allen-Cahn equation and extended to vector-valued equations. Some numerical experiments are conducted to verify the theoretical results.
Lagrange multiplier SAV approach for dissipative/conservative systems
Jie SHEN
Purdue University
Abstract: We present in this talk the Lagrange multiplier SAV approach for dissipative/conservative systems. Unlike the original SAV approach, the Lagrange multiplier SAV approach can dissipate the original energy and does not require that the nonlinear potential part of the energy is bounded from below. Furthermore, it can easily handle dissipative/conservative systems with additional global constraints. We shall present applications of the Lagrange multiplier SAV approach to several typical examples of dissipative and conservative systems as well as to minimization problems
Higher order energy stable ETD based methods for gradient flows
Xiaoming Wang
Southern University of Science and Technology
Abstract: Many natural and engineering problems follow gradient flow structures in the sense that systems evolve to decrease certain energy. The dynamics of most of these gradient systems are complicated and hence numerical methods are called for. There are at least two desirable features for numerical algorithms for gradient flows with long evolution process: efficient higher oder in time, and long-time stability. We present a class of efficient higher-order energy stable methods for a class of gradient flows based on the exponential time differencing (ETD) method combined with multi-step methods and regularization. As a specific example, we present a third order ETD based scheme for thin film epitaxial growth model together with numerical results establishing the convergence and stability of the scheme, and the ability of the scheme to capture long-time scaling properties of the system.
Reduced-order methods of phase field modelling with Auxiliary Variable approach and SEM
Chuanju XU
Xiamen University
Abstract: In this talk we will discuss a variant of phase-field modelling derived from an energy variational formulation. Different models are derived and efficient numerical methods are proposed and analyzed. In particular, we discuss the algorithms based on spectral element method for the spatial discretization and stabilized ADI or auxiliary variable approach for the temporal discretization. Reduced order methods are then developed using by proper orthogonal decomposition and discrete empirical interpolation to improve the computational efficiency.
Efficient, accurate and energy stable discontinuous Galerkin methods for phase field models of two-phase incompressible flows
Yan XU
University of Science and Technology of China
Abstract: The goal of this paper is to propose fully discrete discontinuous Galerkin (DG) finite element methods for the phase field models of two-phase incompressible flows, which are shown to be unconditionally energy stable. In details, using the convex splitting principle, we first construct a first order scheme and a second order Crank-Nicolson scheme for time discretizations. The proposed schemes are shown to be unconditionally energy stable. Then, using the invariant energy quadratization approach, we develop a novel linear and decoupled first and second order scheme, which is easy to implement and energy stable. In addition, a semi-implicit spectral deferred correction method combining with the low scheme is employed to improve the temporal accuracy. Due to the local properties of the DG methods, the resulting algebraic equations at the implicit level is easy to implement and can be solved in an explicit way when it is coupled with iterative methods. In particular, we present efficient and practical multigrid solvers to solve the resulting algebraic equations, which have nearly optimal complexity. Numerical experiments of the accuracy and long time simulations are presented to illustrate the high order accuracy in both time and space, the capability and efficiency of the proposed methods.
Efficient numerical schemes for gradient flow models
Xiaofeng YANG
University of South Carolina
Abstract: 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.
OnsagerNet: learning stable and interpretable dynamics using a generalized Onsager principle
Haijun YU
ICMSEC, Chinese Academy of Sciences
Abstract: Onsager principle is a basic tool to derive mathematical models
for complex systems in fluid dynamics, materials science and biological science. However,there are two challenges in using Onsager principle to build quantitatively accurate models: how to find the generalized coordinates, and how to determine large amounts of coefficients in the models. We propose a systematic method for learning stable and interpretable low-dimensional dynamical models by a combination of a generalized Onsager principle and machine learning. The learned dynamics are autonomous ordinary differential equations parameterized by RePU neural networks that retain clear physical structure information, such as free energy, diffusion, conservative interaction and external force. The neural network representations for the hidden dynamics are trained by using embedded Runge-Kutta method. For high dimensional problems with a low dimensional slow manifold, an autoencoder with isometric regularization is introduced to find generalized coordinates on which we learn the generalized Onsager dynamics. Our method exhibits clear advantages in benchmark problems of learning ordinary differential equations, such as nonlinear Langevin dynamics and the Lorenz system. We further apply this method to study Rayleigh-Benard convection and learn Lorenz-like low dimensional autonomous models that capture both qualitative and quantitative properties of the underlying dynamics. This is a joint work with: Weinan E(Princeton), Qianxiao Li(NUS) and Xinyuan Tian(AMSS).
