A discontinuous Galerkin method for one-dimensional time-dependent nonlocal diffusion problems

李晓，北京计算科学研究中心

Title and Abstract

Numerical Methods for Aerosol Dynamic Equations

崔明

北京工业大学

E-mail address: mingcui@bjut.edu.cn

Abstract: Aerosol modeling is very important to study the behavior of aerosol dynamics in atmospheric environment. In this talk, we consider numerical methods for the nonlinear aerosol dynamic equations on time and particle size. The finite volume element methods based on the linear interpolation and Hermite interpolation are provided to approximate the aerosol dynamic equation where the condensation and removal processes are considered. Numerical examples are provided to show the efficiency of these numerical methods.

Kernel free boundary free method for Stokes interface problems and its maximum error analysis

董海霞

北京计算科学研究中心

E-mail address: hxdong@csrc.ac.cn

Abstract: This talk presents a fast kernel-free boundary integral (KFBI) method via finite difference scheme for the Stokes interface problem on a staggered grid. The idea in our approach is to reformulate the Stokes interface problem with discontinuous viscosity as a boundary integral equation. The most important feature of the proposed method is that we do not directly calculate the integrals, but reinterpret them to an equivalent Stokes interface problem with continuous viscosity. In addition, the equivalent Stokes interface is discretized by the well-known marker and cell (MAC) scheme via finite difference method then solved by a fast Fourier transform (FFT) solvers. Furthermore, we provide rigorous error analysis to show that our method can achieve a second-order convergence rate for the approximation of the velocity and its gradient in maximum norm even if the truncation error is O(h) near the interface Γ. Numerical results are given to demonstrated the efficiency and accuracy of the proposed method.

Continuous Galerkin Solutions for Delay Differential Equations of Pantograph Type

黄秋梅

北京工业大学

E-mail address: qmhuang@bjut.edu.cn

 In this report, we consider the continuous Galerkin method to solve the delay differential equations of pantograph type under uniform and quasi-graded mesh. The global convergence and the local superconvergence results are obtained. Based on the local superconvergence results, several postprocessing techniques to accelerate the global convergence are proposed. The theoretical expectations are confirmed by numerical experiments.

A General tensor based model for active liquid crystals

纪光华

北京师范大学

E-mail address: ghji@bnu.edu.cn

Abstract: We use the generalized Onsager principle to derive a general continuum model for active liquid crystals. This model can be used for both polar and apolar active liquid crystals. Then, we focus on an apolar active liquid crystal model in which the polarity vector vanishes. We study spontaneous flows in infinite domain or a finite domain with periodic boundary conditions and obtain the critical value for the onset of spontaneous flows for perturbations parallel and perpendicular to the base steady state solution, respectively. Two effective parameter combinations are identified that dictates the formation of spontaneous flows with the boundary conditions. Finally, we investigate the onset of spontaneous flows in a channel flows with 1D spatial inhomogeneity. A critical value exists for z.

动力学观点下的癌症演变模型

雷锦誌

清华大学

E-mail address: jzlei@mail.tsinghua.edu.cn

摘要：本报告将介绍关于癌症癌变的动力学模型的几方面工作，主要围绕从动力学观点下对癌症演变过程的不同尺度变化进行数学建模，分别介绍关于表观遗传变化、干细胞增殖的数学模型、癌症演变的动力学等相关内容。（我会重点讲一下我们最近在 Cancer Research 发表的论文。）

Discrete maximum principles of the exponential time differencing schemes for nonlocal Allen-Cahn equations

李晓

北京计算科学研究中心

 E-mail address: xiaoli@csrc.ac.cn

Abstract: In this talk, we present the exponential time differencing (ETD) schemes for solving the nonlocal Allen-Cahn (NAC) equation, a variation of the classic Allen-Cahn equation by replacing the Laplacian by a parameterized nonlocal diffusion operator. Since the solutions to the NAC equation satisfy the maximum principle, numerical approximations preserving the maximum principle in the discrete sense are highly desirable at both physical and mathematical levels. Our numerical schemes are obtained by using the quadrature-based finite difference method for the spatial discretization and applying the ETD-based approximations on the temporal integrations. We establish the discrete maximum principles, and then obtain the energy stability and the maximum-norm error estimates in the fully discrete sense. In addition, we also prove the asymptotic compatibility of the proposed schemes, which implies the robustness of the numerical approximations to the NAC equation. The convergence rates are verified numerically with respect to the discretization and the nonlocal parameters. A further numerical investigation is carried out for the steady state solutions on the relationship between the discontinuities and the nonlocal parameters.

A discontinuous Galerkin method for one-dimensional time-dependent nonlocal diffusion problems

卢键方

北京计算科学研究中心

E-mail address: jflu@csrc.ac.cn

Abstract: In this talk, we introduce a discontinuous Galerkin (DG) method for solving the nonlocal diffusion (ND) problems. The key ingredient of the method is the introduction of an auxiliary variable, analogous to the classic local discontinuous Galerkin (LDG) method but with some nonlocal extensions. Theoretical analysis shows that the proposed semi-discrete DG scheme is $L^2$-stable, convergent and in particular asymptotically compatible. Various numerical tests are provided to demonstrate the effectiveness and robustness of our method. 

Thermodynamical effects of high resolution schemes for compressible fluid dynamics
 汪玥

北京应用物理与计算数学研究所

E-mail address: wang_yue@iapcm.ac.cn

Abstract: One of the fundamental differences of compressible fluid flows from incompressible
fluid flows is the involvement of thermodynamics. This difference should be manifested in the design of numerical schemes. Unfortunately, the role of entropy, expressing irreversiblity, is often neglected even though the entropy inequality, as a conceptual derivative, is verified for some first order schemes. In this talk, we refine the GRP solver to illustrate how the thermodynamical variation is integrated into the design of high resolution methods for compressible fluid flows and demonstrate numerically the importance of thermodynamic effects in the resolution of strong waves. As a by-product, we show that the GRP solver works for generic equations of state, and is independent of technical arguments.

A positivity-preserving, energy stable numerical  scheme for the Cahn-Hilliard equation with logarithmic potential

Cheng Wang (王成) 
University of Massachusetts Dartmouth
E-mail address: cwang1@umassd.edu

Abstract: The Cahn-Hilliard model with logarithmic potential is considered, in which the key difficulty has always been associated with the singularity of the logarithmic terms. An energy stable

finite difference scheme, which implicitly treats the logarithmic terms, is proposed and analyzed in this talk. In particular, how to ensure the positivity of the logarithmic arguments, so that the

numerical scheme is well-defined at a point-wise level, has been a long-standing mathematical challenge. It is proved that, given any numerical solution with a fixed bound at the previous time

step, there exists a unique numerical solution that satisfies the given bound (-1,1) at a point-wise level. As a result, the numerical scheme is proven to be well-defined, and the unique solvability and energy stability could be established with the help of convexity analysis. In addition, an optimal rate convergence analysis could be appropriately established. Some numerical results are also presented in the talk.

A Multilevel Correction Adaptive Finite Element Method for Kohn-Sham Equation.

徐飞

北京工业大学

E-mail address: fxu@bjut.edu.cn

Abstract：An adaptive finite element method will be introduced for solving Kohn-Sham equation with the multilevel correction technique. In the method, the Kohn-Sham equation is solved on a fixed and appropriately coarse mesh with the finite element method in which the finite element space is kept improving by solving the derived boundary value problems on a series of adaptively and successively refined meshes. A main feature of the method is that solving large scale Kohn-Sham system is avoided effectively, and solving the derived boundary value problems can be handled efficiently by classical methods such as the multigrid method. Hence, the significant acceleration can be obtained on solving Kohn-Sham equation with the proposed multilevel correction technique.

Story on the Ericksen-Leslie system of liquid crystal with and without the penalty function张辉

北京师范大学

E-mail address: hzhang@bnu.edu.cn

Abstract: Here we will talk about the story the Ericksen-Leslie system of liquid crystal with and without the penalty function. What is the advantage and disadvantage of the model with and without the penalty function, including the different numerical approach and phenomena. We present the development and interaction of the defects. These results are partly consistent with the observation from the experiments. Thus this scheme illustrates, to some extent, the kinematic effects of the defects.

Fully Discrete Second-order Linear Schemes for Hydrodynamic Phase Field Models of Binary Viscous Fluid Flows with Variable Densities

 Jia Zhao

Utah State University

 E-mail address: jia.zhao@usu.edu

Abstract: In this talk, we present spatial-temporally second-order, energy stable numerical schemes for two classes of hydrodynamic phase field models of binary viscous fluid mixtures of different densities. One is quasi-incompressible while the other is incompressible. We introduce a novel invariant energy qudratization (IEQ) technique to arrive at fully discrete linear schemes, where in each time step only a linear system needs to be solved. These schemes are then proved to be unconditionally energy stable rigorously so that a large time step is plausible. Both spatial and temporal mesh refinements are conducted to illustrate the second order accuracy of the schemes. Several Numerical examples and conducted, and predictions by the two fluid mixture models are compared and discussed. As a conclusion, we believe the quasi-incompressible model is more reliable than the incompressible one.

This is a joint work with Yuezheng Gong (Nanjing University of Aeronautics and Astronautics, China), Xiaogang Yang (Wuhan Institute of Technology, China) and Qi Wang (Beijing Computational Science Research Center, China).

周期性血小板减少症的动力学研究

诸葛昌靖

北京工业大学

E-mail address: zhuge@bjut.edu.cn

摘要：周期性血小板减少症(Cyclical thrombocytopenia, CT)是一种造血系统疾病, 其特征为血小板的数量呈现周期性振荡. 在多数病例中, 除血小板外其他血细胞的数量保持稳定, 但是最近的一些CT病例中也发现了嗜中性粒细胞数量的明显振荡. 我们利用先前已经建立的造血系统的数学模型研究了这一现象, 通过改变控制参数, 我们得到了不同的振荡模式. 我们的结果显示血小板的成熟时间和造血干细胞的分化率在产生这些不同的振荡模式中发挥重要作用. 我们研究了不同的振荡模式产生的机制, 说明了血小板模块发生的病变引起其他血细胞数量振荡的可能机制. 此外, 通过分岔分析, 我们发现了系统中存在的双稳态. 这些结果提示, 血细胞数量动力学的历史轨迹对于治疗CT具有重要意义.

Local Information about lodging, lecture hall and local transportation:

当地信息（住宿、会议地点、交通）：

Hotel Information for Reference  (住宿参考)

▼北京友谊宾馆 

    ☆  中关村南大街1号 

    ☆  010-68498888

 ▼ 汇贤大厦 

    ☆  中关村大街59号（人民大学校内，近人民大学西门） 

    ☆  010-62516038

 ▼ 燕山大酒店（中关村大街店） 

    ☆  中关村大街甲38号（近人民大学东门） 

    ☆  010-62563388

▼ 如家快捷酒店（北京苏州桥店） 

   ☆  中关村南大街1号 

    ☆  010-68498888

 ▼ 汉庭（北京中关村人民大学店）

    ☆  三义庙新华服装厂大华衬衫厂宿舍9号楼（近人民大学东门）

    ☆  010-82656868

▼ 7天连锁酒店(中关村人民大学店) 3.0 （1人评价） 人均：¥287

☆  北京市海淀区苏州街49-2号（海淀南路与苏州街交叉口南行100米西侧）

☆  01062577877|01088411855 

▼ 东方凯德华快捷酒店

☆  北京市海淀区海淀南路9号(海淀医院西)

☆  010-62556666;010-82619998

1.       Location of lecture hall: 110, Guoxue Building  (国学馆110教室), Renmin University of China (中国人民大学国学馆110教室, 图书馆西侧)

2.       Local transportation （To Yanshan Hotel/ East Gate of Renmin University）

(1)     The Capital International Airport首都国际机场

Subway: Take the airport express (fast track), get off at Sanyuan Bridge station, then transfer to subway line 10, next, get off at Haidian Huangzhuang station, and transfer to subway line 4 (Tiangongyuan direction), finally get off at Renmin University Station (exit A1) and walk to Yanshan Hotel/ East gate of Renmin University.

地铁：乘坐机场快轨（三元桥方向），在三元桥站下车，换乘地铁十号线，到海淀黄庄站换乘地铁4号线，到人民大学站下车（A1出口），步行至燕山大酒店或中国人民大学东门。

Public bus: Take the airport bus to the direction of Gongzhufen, and get off at Youyi Hotel stop, and then walk to the east gate of Renmin University.

地面公交: 乘坐机场大巴公主坟线，在友谊宾馆站下车，步行至中国人民大学东门。

By Taxis: this costs approximately 100 RMB. Please show the driver:

     请带我去人民大学东门对面的燕山大酒店 （please take me to Yanshan Hotel）

出租：打车约100元人民币。

(2)    Beijing Railway Station北京站

Subway: Take subway line 2 (internal) and get off at Xizhimen station, and then transfer to subway line 4 to the direction of North Anhe Bridge, get off at Renmin University station, exit A1 , and walk to Yanshan Hotel / East gate of Renmin University. By taxis, it’ll cost about 60 RMB.

地铁：乘坐地铁2号线（内环），在西直门站下车，换乘地铁4号线（安河桥北）方向，在人民大学站下车（A1口出），步行至燕山大酒店或中国人民大学东门。

出租车：打车费用约60元人民币。

(3)    Beijing West Railway Station北京西站：

Subway: Take subway line 9 and get off at the National library station, and then transfer to subway line 4 in the direction of North Anhe Bidge, get off at Renmin University station from the A1 gate, and walk to the Yanshan Hotel / east gate of Renmin University.

地铁：出站后乘坐地铁9号线（国家图书馆方向），在国家图书馆站下车，换乘地铁4号线（安河桥北方向），在人民大学站下车，（A1口出）。步行至燕山大酒店或中国人民大学东门。

Public bus: ① Take the No. 374 bus, get off at Wanquanzhuang stop, and walk to the west gate of Renmin University. ② Take the special No. 18 or 6 bus, get off at Renmin University stop, and then walk to the east gate of Renmin University. By taxis; this costs approximately 35 RMB.

公交：①在北京西站公交站乘坐374路公交，在万泉庄站下车，步行至中国人米大学西门。②在北京西站公交站乘坐特18路或特6路，在人民大学站下车，步行至燕山大酒店或中国人民大学东门。

出租：打车费用约为35元人民币。

(4)    Beijing South Railway Station北京南站：

Subway: Take subway line 4 in the direction of North Anhe Bidge, get off at Renmin University station from the A1 gate, and walk to Renmin University.

地铁：乘坐地铁4号线（安河桥北方向），在人民大学站下车，（A1口出），步行至中国人民大学东门。

Public bus: Take the special No. 5 bus, get off at Wanquanzhuang stop, and then walk to the west gate of Renmin University. By taxis; this costs about 60 RMB.

公交：在北京南站南广场乘坐特5路公交，在万泉庄站下车，步行至中国人民大学西门。               出租：打车费用约为60元人民币。

人大校内Wi-Fi使用办法

(To use Wi-Fi on RUC campus, please ask the organizers for help)

Step 1: 开启Wi-Fi接收，人大Wi-Fi名称前缀是RUC_，打开浏览器，在地址栏中输入go.ruc.edu.cn并点击回车

Step 2：点击 “手机注册”（左上图是电脑登录页面，右上图是手机登录页面）

Step 3：填写手机号，姓名，单位，验证码后点击“下一步”，手机会收到一条4位验证码的短信，点击右上角的“×”回到Step 1的页面，在“身份认证”第一栏输入手机号，第二栏输入刚收到的4位验证码，点击“登录”即连入Wi-Fi。