2019年学术委员会会议

暨微分方程理论与计算前沿问题研讨会

工程建模与科学计算湖北省重点实验室自2016年12月获得认定以来已运行近2年半，为了规范和科学地建设运行实验室，加快推进实验室的各项工作，实验室将于2019年6月21日-24日在华中科技大学召开2019年学术委员会会议暨微分方程与计算前沿问题研讨会,敬请光临。

会议日程表

6月21日下午

注册报到（地点：华中科技大学国际学术交流中心八号楼大厅）

6月22日上午

8：30-8：45：开幕式（地点：科技楼702）

主持人：张诚坚

程序：1.数学与统计学院院长吴军教授致欢迎词

2.拍照

6月22日下午

地点：科技楼702

6月23日上午

地点：702会议室

6月23日下午

地点：702会议室

6月24日上午

会议闭幕，来宾离会

学术报告摘要

不可压欧拉方程定常涡解的存在性、唯一性和稳定性

曹道民

中国科学院数学与系统科学研究院广州大学

报告人将报告近年来在不可压欧拉方程定常涡解方面的研究，特别地要介绍在涡补丁解（vortex patch)的存在性和唯一性方面的结果。报告人将讲述涡补丁解的存在唯一性与Kirchhoff - Routh函数临界点之间的联系，而Kirchhoff - Routh临界点又与方程所在区域的几何性质密切相关。此外，报告还将介绍以涡补丁解为初值的解的稳定性的一些结果。

Two-stage fourth-order accurate time discretizations for 1D

and 2D special relativistic hydrodynamics

汤华中

北京大学数学科学学院湘潭大学

This paper studies the two-stage fourth-order accurate time discretization [2] and applies it to special relativistic hydrodynamical equations. It is shown that new two-stage fourth-order accurate time discretizations can be proposed. With the aid of the direct Eulerian GRP (generalized Riemann problem) methods and the analytical resolution of the local \quasi 1D" GRP, the two-stage fourth-order accurate time discretizations are successfully implemented for the 1D and 2D special relativistic hydrodynamical equations. Several numerical experiments demonstrate the performance and accuracy as well as robustness of our schemes.

Recent Development on Superconvergence Theory in

Computational Mathematics

张智民

北京计算科学研究中心

Superconvergence phenomenon is well understood for the h-version finite element method and researchers in this old field have accumulated a vast literature during the past 40 years. However, the relevant systematic study for discontinuous Galerkin, finite volume, and spectral methods is lacking. We believe that the scientific community would also benefit from the study of superconvergence phenomenon of those methods. Recently, some efforts have been made to expand the territory of the superconvergence. In this talk, I will summarize some recent development on superconvergence study for these methods. At the same time, some current issues and un-solved problems will also be addressed.

离散动力系统的统计性质

周爱辉

中国科学院数学与系统科学研究院

我们将通过简单的例子说明，不同视角下的离散动力系统的渐近性态会不同。不少确定性系统的渐近性态常表现为混沌现象,无法预测。然而,如果用统计的观点来看混沌动力系统,确定性意义下的混沌在统计意义下又常具有正规性。

Most Probable Transition Pathways in Stochastic

Dynamical Systems

段金桥

华中科技大学

Dynamical systems arising in engineering and science are often subject to random fluctuations. The noisy fluctuations may be Gaussian or non-Gaussian, which are modeled by Brownian motion or α-stable Levy motion, respectively. Non-Gaussianity of the noise manifests as nonlocality at a “macroscopic” level. Stochastic dynamical systems with non-Gaussian noise (modeled by α-stable Levy motion) have attracted a lot of attention recently. Thenon-Gaussianity indexα is a significant indicator for various dynamical behaviors.

The speaker will present recent work on most probable transition pathways between metastable states, for stochastic dynamical systems with non-Gaussian Levy noise. This is joint work with Ying Chao.

Unconditionally convergent difference schemes for

the fractional Ginzburg-Landau equation

黄乘明

华中科技大学

In this talk, we propose and analyze two efficient difference schemes for the nonlinear complex Ginzburg-Landau equation involving fractional Laplacian. The first one is a fully implicitscheme of order O(\tau^2+h^2) with time step\tau and mesh size h. The second one is an implicit-explicit scheme of order O(\tau^2+h^4). Both schemes are proved to beunconditionally convergent, in the sense that the time step \tau does not need to be related to the spatial mesh size h when both of them go to zero. Numerical tests are provided to confirm the accuracy and efficiency of the two schemes.

Geometric singular perturbation analysis to variants of Camassa-Holm equation

李骥

华中科技大学

In the first part, We introduce the Camassa-Holm equation as one of shallow water model, and compare it with other models. In the second part, we apply geometric singular perturbation theory to show the persistence of solitary wave under small distributed delaying environment. In the third part, we give an artificial inclined thin film model and analyze the fate of solitary wave as well as peakons.

Generalized lattice Boltzmann method：

Modeling, analysis, elements, and applications

施保昌

华中科技大学

In this talk, we first present a unified framework for the modelling of generalized lattice Boltzmann method (GLBM). We then conduct a comparison of the four popular analysis methods (Chapman-Enskog analysis, Maxwell iteration, direct Taylor expansion and recurrence equations approaches) that have been used to obtain the macroscopic Navier-Stokes equations and nonlinear convection-diffusion equations from the GLBM, and show that from mathematical point of view, these four analysis methods are equivalent to each other. Finally, we give some elements that are needed in the implementation of the GLBM, and also find that some available LB modelscan be obtained from this GLBM.

Asymptotic Properties of Delay Differential Systems with Wideband Noise Perturbations

吴付科

华中科技大学

Because a wideband noise is easily realized in applications and because it well approximates a white noise, this work focuses on delay differential equations subject to wideband noise perturbations. By using perturbed test function methods combined with martingale techniques, this paper shows that when the small parameter tends to zero, the underlying process converges to a limit that is governed by solutions of a stochastic delay differential equation. It is noticeable that this limit stochastic differential equation not only depends on the delay term, but also the second delay. Not only are the results interesting from a mathematical point of view, but also they will be of great utility to a wide range of problems in control and optimization problems.

全局渐近稳定性的Markus-Yamabe猜想以及多项式的

Jacobi猜想

杨晓松

华中科技大学

报告针对动力系统平衡点的全局渐近稳定性问题，介绍了Markus-Yamabe猜想及其国际上这方面的相关工作，以及在反馈镇定方面的应用。并讨论了该猜想的研究和代数几何中的著名的Jacobi猜想之间的关系、相关工作以及背后的轶事。

Generalized Störmer–Cowell Methods for

Nonlinear BoundaryValue Problems of Second-Order Delay-Integro-DifferentialEquations

张诚坚

华中科技大学

This talkdeals with the numerical solutions of nonlinear boundary valueproblems (BVPs) of second-order delay-integro-differential equations. The generalized Störmer-Cowell methods (GSCMs), combined with the compound quadrature rules, areextended to solve this class of BVPs. It is proved under some suitable conditions that theextended GSCMs are uniquely solvable, stable and convergent of order min{p, q}, wherep, q are consistent order of the GSCMs and convergent order of the compound quadraturerules, respectively. Several numerical examples are presented to illustrate the proposed methodsand their theoretical results.Moreover, a numerical comparison with the existed methodsis also given, which shows that the extended GSCMs are comparable in numerical precisionand computational cost.

青年创新基金摘要

微磁学方程的高效数值求解

高华东

本项目主要研究微磁材料模型中的非线性偏微分方程的高效数值计算。我们主要关注微磁学模型中的朗道-利夫希兹(Landau-Lifshitz)方程。它在工业生产中具有重要的应用价值。其数值计算的难点在于方程的强非线性以及约束|m|=1。本项目将发展满足|m_h|=1的有限元格式。我们将严格分析这些新格式的稳定性和收敛性。我们期望这个项目的研究结果能为微磁学模型的高效数值计算求解提供新的计算方法。

分数阶偏微分方程的自适应多区域谱方法

刘飞

分数阶偏微分方程广泛应用于模拟反常扩散等物理系统。由于分数阶微分、积分算子的非局部、奇异性，模型解通常是非光滑函数，对比于整数阶偏微分方程，分数阶方程数值解的计算尤其困难。不同于带权的雅可比多项式作为基函数，在单个区域上的谱方法，我们对左、右黎曼-刘维尔分数阶导数，卡普托分数阶导数构造多区域谱微分矩阵。为处理数值解的奇异性，我们采用自适应的h-p网格加密方法，以及自适应移动网格方法，数值计算非线性的分数阶偏微分方程，如分数阶Burgers方程等。对比于存在的分数阶方程数值方法，我们有更好的数值精度，很好地刻画了解的奇异性和Burgers方程陡峭传播的分辨率。

电路中随机微分代数模型的新型高效仿真算法

覃婷婷

电路分析中常有随机因素的干扰，建模时亦受代数约束条件的限制，因此随机微分代数系统模型能更好地模拟实际情况。然而该类模型的高复杂度使得其数值模拟尤为困难，现有算法具有精度低或计算量大的局限。本课题拟针对电路中的随机微分代数系统模型，设计具有更高精度和低计算量的新型高效数值仿真算法，并分析算法的相关数值性质，为新开发的算法提供理论保障。利用仿真实验，验证算法的效率和可靠性，并根据仿真结果分析随机因素及约束条件对电路分析模型的具体影响。

高维浅水波模型若干问题的研究

严凯

孤立子与波的破裂是浅水波中两个最基本的现象。著名的Camassa-Holm (CH)方程及其双分量系统是描述它们的一维非线性色散浅水波模型。本项目主要研究高维浅水波系统。综合使用Bony仿积分解、交换子估计、Littlewood-Paley分解以及能量方法，我们拟在Triebel-Lizorkin空间框架下建立传输方程的相关理论。再据此建立上述系统Cauchy问题的局部适定性及其强解的爆破机制。进一步，我们将借助系统自身结构与恰当的守恒律构造其整体强解与爆破强解，并研究其整体弱解的存在唯一性。本项目的研究将加深我们对浅水波的理解，在理论与实际应用上都具有重要意义。

带奇性Cucker-Smale方程的群体行为和平均场极限

张雄韬

复杂多体系统的动力学行为研究正在高速发展之中，与社会科学、工程控制以及生命科学等学科都密切相关。Cucker-Smale (CS)模型正是研究多体系统集体行为的非常重要和经典的模型之一。目前的研究主要集中在正则交互作用的情形，而在很多情形下，类似于库仑力等带奇性的交互作用更具有现实意义，能很好地与物理、生物等学科的需求相兼容。本项针对带奇性交互作用的CS模型所做的研究包括：1.严格证明平均场极限，从而论证围观模型与介观模型的统一性；2.研究Kinetic CS模型的正则性；3.研究CS模型多集群的涌现行为。本课题是多体系统动力学行为研究的一个自然深入的发展，对于多体系统中集体行为涌现的研究有重要的学术价值，也为样本数量特别巨大的系统仿真模拟提供了有效的方法和理论基础。