International Workshop on Numerical Method, Scientific Computing and Applications

2019710日—712日，山东烟台）

会议安排

1.      报到时间：201971014:0022:00

2.      报到地点：烟台丽景海湾酒店（山东省烟台市莱山区枫林路25号）

3.      会议费用：本次会议不收取会务费，食宿统一安排，费用自理。

会议组织

Xiaofeng Yang (杨霄锋)   University of South Carolina

Chuanjun Chen  (陈传军烟台大学数学与信息科学学院

会议联系人

Guanyu Xue (薛冠宇秘书)  Email: gyxue@ytu.edu.cn, Tel: +86-13685350531

Chuanjun Chen (陈传军)  Email: cjchen@ytu.edu.cn,  Tel: +86-13723947906

 2019年7月11日上午 08:30—09:00 开幕式及会议合影 Morning Session   1  Chair: 杨霄锋 09:00—09:30 鞠立力（University of   South Carolina）Conservative   Explicit Local Time-stepping Schemes for the Shallow Water Equations 09:30—10:00 蔡勇勇（北京计算科学研究中心）Ground states   of spinor Bose-Einstein condensates 10:00—10:30 潘克家（中南大学）求解大规模问题的外推多网格算法 10:30—10:50 休息coffee Morning Session   2  Chair: 鞠立力 10:50—11:20 谢小平（四川大学）Efficient   algorithms for time fractional diffusion and wave equations 11:20—11:50 王疆兴（湖南师范大学）Error analysis   of fully discrete HDG method for Allen-Cahn equation 11:50—12:20 王坤（重庆大学）Efficient   Discrete Methods for Solving Time Fractional Diffusion Equations 午餐

 2019年7月11日下午 Afternoon   Session 1  Chair: 蔡勇勇 14:00—14:30 杨霄锋（University of   South Carolina）Efficient   schemes with unconditionally energy stabilities for a new modified phase   field surfactant model 14:30—15:00 张辉（北京师范大学）Topological   defects in two-dimensional liquid crystals confined by a box 15:00—15:30 岳兴业（苏州大学）Numerical   methods on the population genetic drift problems 15:30—16:00 龚跃政（南京航空航天大学）Preserving-structure   Algorithms for Hydrodynamic Phase Field Models of Binary Viscous Fluid Flows   with Different Densities 16:00—16:20 休息coffee Afternoon   Session 2  Chair: 张辉 16:20—16:50 张翔雄（Purdue   University）Monotonicity   and discrete maximum principle in high order accurate schemes for diffusion   operators 16:50—17:20 赵佳（Utah State University）Structure-Preserving   Numerical Approximations for Thermodynamically Consistent Models 17:20—17:50 仲杏慧（浙江大学）An Efficient   WENO Limiter for Discontinuous Galerkin Transport Scheme on the Cubed Sphere 17:50—18:20 朱立永（北京航空航天大学）GPU-accelerated   Runge-Kutta exponential time difference methods for parabolic equations 晚宴

 2019年7月12日 全天自由交流

（按姓氏字母顺序排列）

Ground states of spinor Bose-Einstein condensates

Abstract: The remarkable experimental achievement of Bose-Einstein condensation (BEC) in 1995 has drawn significant research interests in understanding the ground states and dynamics of trapped cold atoms. Different from the single component BEC, spinor BEC possesses the spin degree of freedom and exhibits rich phenomenon. In the talk, we will introduce some mathematical results for ground states of spin-1,2 BECs, and a practical imaginary time propagation method for numerical simulation with several different projection strategies

Preserving-structure Algorithms for Hydrodynamic Phase Field Models of Binary Viscous Fluid Flows with Different Densities

Abstract: Hydrodynamic phase field models of binary viscous fluid flows can be derived by the generalized Onsager principle and thus possess an important mathematical structure, i.e. the energy dissipation law. In this talk, we develop the preserving-structure algorithms for the two-phase hydrodynamic phase field models, which are proved rigorously to preserve the discrete energy dissipation law. Some numerical examples are presented to show the effectiveness of our proposed schemes.

Conservative Explicit Local Time-stepping Schemes for the Shallow Water Equations

University of South Carolina

Abstract: In this talk we present explicit local time-stepping (LTS) schemes with second and third order accuracy for the shallow water equations. The system is discretized in space by a C-grid staggering method, namely the TRiSK scheme adopted in MPAS-Ocean, a global ocean model with the capability of resolving multiple resolutions within a single simulation. The time integration is designed based on the strong stability preserving Runge-Kutta (SSP-RK) methods, but different time step sizes can be used in different regions of the domain through the coupling of coarse-fine time discretizations on the interface, and are only restricted by respective local CFL conditions. The proposed LTS schemes are of predictor-corrector type in which the predictors are constructed based on Taylor series expansions and SSP-RK stepping algorithms. The schemes preserve some important physical quantities in the discrete sense, such as exact conservation of the mass and potential vorticity and conservation of the total energy within time truncation errors. Moreover, they inherit the natural parallelism of the original explicit global time-stepping schemes. Extensive numerical tests are presented to illustrate the performance of the proposed algorithms.

Error analysis of fully discrete HDG method for Allen-Cahn equation

Hunan Normal university

Abstract: In this talk we present a linearized scheme that combine Invariant Energy Quadratization (IEQ) approach and the hybridizable discontinuous Galerkin method (HDG) for solving the time-dependent nonlinear Allen-Cahn equation. The optimal $L^2$ error estimates for the HDG approximations to the solution and its gradient are established with no restriction on time step size.     Numerical experiments illustrate that the order of convergence obtained in the theoretical analysis is sharp. This talk is based a joint work with Xiaofeng Yang.

Efficient Discrete Methods for Solving Time Fractional Diffusion Equations

College of Mathematics and Statistics, Chongqing University

Abstract: The differential equation with a fractional time derivative of order $\alpha\in(0,1)$ has gained much attention. Due to the existences of the memory and the singularity, to design efficient methods is necessary when solving this problem. In this talk, two numerical methods are investigated. The first one is a fast algorithm with almost optimum memory, which is based on a nonuniform split of the interval $[0,t_n]$ and a polynomial approximation of the kernel function $(1-\tau)^{-\alpha}$, and reduces both the storage requirement and computational cost from $O(n)$ to $O(\log n)$, but keeps the same convergence rate as that of the corresponding direct method. The second one is a new discrete scheme focusing on the weak singularity near the initial time $t=0$, which is based on the novel proposed nonuniform meshes (the tanh meshes) by using the $L1$ formula. The scheme is proved to be unconditionally stable and reach the ideal convergence rate $2-\alpha$ by suitably choosing the parameter. Some numerical tests are carried out to confirm the efficiency of two methods. This is a joint work with Dr. Jizu Huang, Jiali Zhang and Dr. Xin Wang.

Efficient algorithms for time fractional diffusion and wave equations

School of Mathematics, Sichuan University, Chengdu 610064, China

Abstract: We consider several numerical methods for time fractional diffusion and wave problems. The regularity of the weak solutions to the problems with nonsmooth data are investigated. For the time fractional diffusion problems, we analyze a time-stepping finite element method and a discontinuous Galerkin method. For the time fractional wave problems, we analyze three numerical methods, i.e. a time-spectral finite element method, a space-time finite element method, and a Petrov-Galerkin method. Stability and convergence of the algorithms are derived. Numerical experiments are performed to verify the theoretical results.

This is joint work with Binjie Li and Hao Luo.

Efficient schemes with unconditionally energy stabilities for a new modified phase field surfactant model

University of South Carolina

Abstract: We consider numerical approximations for a new surfactant phase field model. We first reformulate the commonly used kumora’s surfactant model such that the total energy is bounded from below. Furthermore, we combine the recently developed SAV approach with the stabilization technique, where a linear stabilization term is added, which is shown to be crucial to enhance the stability and keep the required accuracy while using large time steps. We further prove the unconditional energy stability of the scheme  rigorously, and present various 2D and 3D numerical simulations to demonstrate the stability and accuracy, numerically.

Numerical methods on the population genetic drift problems

Abstract: Random genetic drift occurs at a single unlinked locus with two or more alleles. The probability density of alleles is governed by a degenerated   Fokker-Planck equation. Due to the degeneration and convection, Dirac singularities will always be developed at boundary as time evolves, which is just the so-called fixation phenomenon. In order to find a complete solution which should keep the conservation of positivity, total probability and expectation, different schemes of FDM, FVM and FEM are tested to solve the equation numerically. We observed that the methods have totally different behaviors. Some of them are stable and keep the conservation of positivity and probability, but fail to keep the expectation. Some of them fails to keep the positivity. Careful analysis is presented to show the reason why one central scheme does work and the others fail. Our study shows that the numerical methods should be carefully chosen and any method with intrinsic numerical viscosity or anti-diffusion must be avoided. Numerical methods for multi-alleles are also discussed.

This is a joint work with Xinfu Chen, Chun Liu, David Waxman and Shixin Xu.

Topological defects in two-dimensional liquid crystals confined by a box

School of Mathematical Sciences, Beijing Normal University, China

Abstract: When a spatially uniform system that displays a liquid-crystal ordering on a two-dimensional surface is confined inside a rectangular box, the liquid crystal direction field develops inhomogeneous textures accompanied by topological defects because of the geometric frustrations. We show that the rich variety of nematic textures and defect patterns found in recent experimental and theoretical studies can be classified by the solutions of the rather fundamental, extended Onsager model. This is critically examined based on the determined free-energies of different defect states, as functions of a few relevant, dimensionless geometric paramets.

Monotonicity and discrete maximum principle in high order accurate schemes for diffusion operators

Purdue University

Abstract: In many applications modeling diffusion, it is desired for numerical schemes to have discrete maximum principle and bound-preserving (or positivity preserving) properties. Monotonicity of numerical schemes is a convenient tool to ensure these properties. For instance, it is well know that second order centered difference and piecewise linear finite element method on triangular meshes for the Laplacian operator has a monotone stiffness matrix, i.e., the inverse of the stiffness matrix has non-negative entries because the stiffness matrix is an M-matrix. Most high order accurate schemes simply do not satisfy the discrete maximum principle. In this talk, I will first review a few known high order schemes satisfying monotonicity for the Laplacian in the literature then present a new result: the finite difference implementation of continuous finite element method with tensor product of quadratic polynomial basis is monotone thus satisfies the discrete maximum principle for the variable coefficient Poisson equation. Such a scheme can be proven to be fourth order accurate. This is the first time that a high order accurate scheme that is proven to satisfy the discrete maximum principle for a variable coefficient diffusion operator. Applications including compressible Navier-Stokes equations will also be discussed.

Structure-Preserving Numerical Approximations for Thermodynamically Consistent Models

Utah State University

Abstract: In this talk, I will first present a general approach for deriving thermodynamically consistent models using the generalized Onsager principle. It turns out many existing models in literature are special cases of the generalized model formulation. Then, I will explain how to utilize the energy quadratization strategy to develop efficient, high-order accurate and structure-preserving numerical approximations for a class of thermodynamically consistent models. Applications of this modeling and numerical paradigm will be discussed.

An Efficient WENO Limiter for Discontinuous Galerkin Transport Scheme on the Cubed Sphere

Abstract: The discontinuous Galerkin (DG) transport scheme is becoming increasingly popular in the atmospheric modeling due to its distinguished features, such as high-order accuracy and high-parallel efficiency. Despite the great advantages, DG schemes may produce unphysical oscillations in approximating transport equations with discontinuous solution structures including strong shocks or sharp gradients. Nonlinear limiters need to be applied to suppress the undesirable oscillations and enhance the numerical stability. It is usually very difficult to design limiters to achieve both high-order accuracy and non-oscillatory properties, and even more challenging for the cubed-sphere geometry. In this paper, a simple and efficient limiter based on the Weighted Essentially Non-Oscillatory (WENO) methodology is incorporated in the DG transport framework on the cubed sphere. The uniform high-order accuracy of the resulting scheme is maintained due to the high order nature of WENO procedures. Unlike the classic WENO limiter, for which the wide halo region may significantly impede parallel efficiency, the simple limiter requires only the information from the nearest neighboring elements without degrading the inherent high-parallel efficiency of the DG scheme. A bound preserving filter can be further coupled in the scheme which guarantees the highly desirable positivity-preserving property for the numerical solution. The resulting scheme is high-order accurate, non-oscillatory, and positivity-preserving for solving transport equations based on the cubed-sphere geometry. Extensive numerical results for several benchmark spherical transport problems are provided to demonstrate good results, both in accuracy and in non-oscillatory performance.

GPU-accelerated Runge-Kutta exponential time difference methods for parabolic equations

School of Mathematics and Systems Science, Beihang University

Abstract: Recently, some novel fast Runge-Kutta exponential time difference (RKETD) methods are developed for solving stiff nonlinear parabolic equations. By employing efficient decompositions of compact spatial difference operators on a regular mesh and FFT-based fast calculations, the methods can obtain low computation cost. However, it is still desired to develop their acceleration version in order to solve large scale realistic problems in three dimensions. In this work, we present a Graphics Processing Unit(GPU) accelerated Runge-Kutta exponential time difference method for parabolic equations base on CUDA. In the proposed method, the cuFFT library is employed to compute FFTs on GPU, while the cuBLAS library is used to realize the basic matrix operations on GPU. The comparison of the numerical results of CUDA-implemented RKETD method on GPU and original RKETD method on CPU demonstrates that the CUDA-implemented RKETD method on GPU can obtain better speedup performance. Numerical experiments demonstrate effectiveness and efficiency of the GPU acceleration for both linear and nonlinear parabolic problems.