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Title and Abstract Shijin Deng Shanghai Jiao Tong University Title: Green's functions of wave equations in ${\mathbb R}_+^n \times {\mathbb R}_+$ Abstract: In this talk we implement the LY algorithm, and introduce the Rayleigh surface wave operators, delayed/advanced mirror images, wave recombinations, and wave cancellations to obtain the Green's functions for the d'Alembert equation with respect to various boundary conditions. We have obtained the complete and simple formula of the Green's functions for the wave equation with the presence of various boundary conditions. We have identified whether a Rayleigh surface wave is active or virtual. Finally, we apply our theorem for the case with virtual Rayleigh surface wave to study the lacunas of the wave equation in 3D with the presence of boundary. ********************************************************* Hao Gao Shanghai Jiao Tong University Title: Accurate Particle Transport in Radiation Therapy Abstract: I will present both the need and our ongoing effort on the accurate particle transport that is an essential prerequisite for the accurate radiation dose delivery to the cancer tumors while sparing the healthy organs, which is particularly true for the state-of-art radiation therapy techniques, such as photon therapy. ********************************************************* Zhongyi Huang Tsinghua University Title: Monotone finite point method for non-equilibrium radiation diffusion equations Abstract: In this talk, we propose the monotone tailored-finite-point method for solving the non-equilibrium radiation diffusion equations. We first give two tailored finite point schemes for the nonlinear parabolic equation in one-dimensional case, and then extend the idea to solve the radiation diffusion problem in 1D as well as 2D. By using variable substitute, our method satisfies the discrete maximum principle automatically, thus preserves the properties of monotonicity and positivity. Numerical results show that our method can capture the sharp front with a coarse mesh size, which is clearly superior to traditional methods. ********************************************************* Ning Jiang Tsinghua University ********************************************************* Song Jiang Institute of Applied Physics and Computational Mathematics Title: Asymptotic Preserving Unified Gas Kinetic Scheme for Grey Radiative Transfer Equations Abstract: The solutions of radiative transport equations can cover both optical thin and optical thick regimes due to the large variation of photon’s mean-free path and its interaction with the material. In the small mean free path limit, the nonlinear time-dependent radiative transfer equations can converge to an equilibrium diffusion equation due to the intensive interaction among radiation and material. In the optical thin limit, the particle free transport mechanism will emerge. In this paper, we are going to develop an accurate and robust asymptotic preserving unified gas kinetic scheme (AP-UGKS) for the grey radiative transfer equations, where the radiation transport equation is coupled with the material thermal energy equation. The current work is based on the UGKS framework for the rarefied gas dynamics [K. Xu and J. Huang, J. Comput. Phys. 229 (2010), 7747-7764], and is an extension of a recent work [L. Mieussens, J. Comput. Phys. 253(2013), 138-156] from a one-dimensional linear radiation transport equation to a nonlinear two-dimensional grey radiative system. The newly developed scheme has the asymptotic preserving (AP) property in the optically thick regime in the capturing of diffusive solution without using a cell size being smaller than the photon’s mean free path, and the scheme can capture the exact solution in the optical thin regime as well. The current scheme is a finite volume method, which is distinguishable from the standard SN-method for radiation transfer equations with ordered computational cells. Numerical tests are presented to validate the current approach. (Joint work with Wenjun Sun and Kun Xu) ********************************************************* 李若 北京大学 题目:一般性动理学方程的模型约化 摘要:动理学方程的模型约化目标是将高维问题转化为低维问题,通过对约化模 型的研究,不但能够对原问题提供更多的理解,而且可以本质提高数值模拟的效 率。由于约化模型是一个具有柯西数据的拟线性系统,其适定性要求系统具有双 曲性。在已经给出的约化模型中,只有很少结果是双曲的从而受到了关注。一些 分析猜测双曲性可能会和模型约化的拟设形式、封闭的方式、热力学第二定律是 否满足、系统的某些守恒性等物理性质有关系。在此处汇报的工作中,我们指出 这些猜测是不正确的。而且,我们给出了一种模型约化的框架,使得可以通过完 全机械的运算,基于任意的拟设和封闭方式,都给出对称双曲的约化模型。可以 看到,文献中已有的双曲模型均可以纳入此框架。作为一个应用,我们给出了一 个具有全局双曲性的 13 矩系统,这是历史上从未有过的新模型。 ********************************************************* ZhiHui Li China Aerodynamics Research and Development Center Title: Computable Modeling of the Boltzmann Transport Equation and Massive Parallel Implementation for Re-entering Aerodynamics Abstract: The Boltzmann equation can describe the gas transport phenomena for the full spectrum of flow regimes and act as the main foundation for the study of complex gas dynamics including spacecraft re-entering Earth’s atmosphere. However, the difficulties encountered in solving the full Boltzmann equation are mainly associated with the nonlinear multidimensional integral nature of the collision term, and an exact solution of the Boltzmann equation is impractical for the analysis of practical complex flows. From the kinetic-molecular theory of gases, numerous statistical or relaxation kinetic model equations resembling various order of moments of the original Boltzmann equation have been put forward. In this work, instead of solving the full Boltzmann equation, it can be indicated that a unified computational modeling based on the Boltzmann equation has been presented in describing flow transport phenomena around complex bodies in various flow regimes, and the theory and computational techniques of a gas-kinetic unified algorithm (GKUA) have been established and used to simulate the reentry aerodynamics from highly rarefied free-molecular one to continuum regimes with the development of massive parallel implementation. Based on the collision relaxation spacing theory of the Boltzmann equation, the unified Boltzmann model equation for describing the complex multi-scale flows covering various flow regimes can be deduced, in which the unified expressions on the molecular collision relaxing parameter and the local equilibrium distribution function are presented by computable modeling of the collision integral of the Boltzmann equation for the full spectrum of flow regimes. The unified expressions are integrated with the macroscopic flow variables, the gas viscosity transport coefficient, the thermodynamic effect, the molecular power law, molecular models, and the flow state controlling parameter from various flow regimes. The gas-kinetic finite difference scheme is constructed to directly solve the discrete velocity distribution functions by using the discrete velocity ordinate (DVO) technique and the unsteady time-splitting method. The discrete velocity numerical integration method is developed to evaluate the macroscopic flow parameters at each point in the physical space. The computing principle of domain decomposition is investigated on the basis of two-phase six-dimensional space of physical space and velocity space, and then the computing technique of parallel domain decomposition in the discrete velocity space is presented. As a result, the gas-kinetic massive parallel algorithm is developed to solve the hypersonic aerothermodynamics covering various flow regimes and the parallel speed-up almost goes up as near-linearity with the increase of the number of processors from 64~32768CPU and processor cores 500~45000 and 3125~112500 at least 88% parallel efficiency. To validate the accuracy and feasibility of the GKUA, the re-entering hypersonic flows past reusable spherical-cone satellite and spacecraft covering the whole of flow regimes are simulated. The computational results are found in good agreement with the related theoretical, DSMC,N-S, and experimental results. The computing practice has confirmed that the present gas-kinetic algorithm probably provides a promising approach for treating practical hypersonic flows during spacecraft re-entry from the gas-kinetic point of view of solving the Boltzmann model equation. ********************************************************* Tiao Lu Peking University Title: Mathematical Analysis and Numerical Simulation of the Wigner Transport Equation Abstract: In this talk, I will introduce the Wigner transport equation (WTE), which can be regarded as a quantum correction of the Boltzmann equation. The WTE has found many applications in many fields, such as nanoscale semiconductor device simulations and quantum optics. But the well-posedness of the stationary Wigner equation with inflow boundary conditions is still an open problem. I will introduce the recent progress on mathematical analysis and numerical simulation of the WTE. My talk will focus on the following three works cooperated with Ruo Li and Zhangpeng Sun at Peking University. First, we proved that the solution of the WTE with inflow boundary conditions is symmetric if the potential is symmetric even if the inflow conditions are asymmetric. Second, we proposed a semi-discretion method and showed that the solution of the semi-discrete WTE goes to the continuous WTE under some conditions as the mesh size of the velocity goes to zero. Third, we proved that the solution of the WTE of a close quantum system must be an even function of velocity. ********************************************************* Zhonghua Qiao Hong Kong Polytechnical University Title: A robust multigrid solver for extended hydrodynamic models in silicon semiconductor Abstract: In this work, we derive a series of extended hydrodynamic models from the Boltzmann transport equation (BTE) by using the Grad moment expansion and globally hyperbolic closed technique for the simulation of semiconductor device. Such models can capture the major features of the solution of BTE, and the solutions recovered from these models converge to the solution of BTE, as the order of the models increase. As a practical application for these models in semiconductor devices, where we are mainly concerned with the stationary solution, we developed a robust multigrid solver for them to further improve the numerical efficiency. Numerical simulations of the carrier transport in a silicon $n^+$-$n$-$n^+$ diode are carried out, to verify the accuracy and convergence of the extended hydrodynamic models. Numerical efficiency of the multigrid solver is also shown by these simulations. ********************************************************* Peng Song Institute of Applied Physics and Computational Mathematics 题目: 二维柱坐标系辐射输运方程的保球对称离散纵标格式 摘要: 在激光间接驱动惯性约束聚变过程中,辐射驱动的靶丸内爆过程直接关系到核燃 料是否能够发生核聚变并持续燃烧释放大量能量,辐射烧蚀、物质界面的不稳定 性发展和物质混合是物理研究和数值模拟的重要问题。传统的二维柱坐标系下辐 射输运方程的离散纵标格式在模拟球对称内爆过程时不能够保持球对称性,并且 数值扰动随着冲击波的聚心反弹而被剧烈放大,给内爆过程的物理分析带来了很 大的困难。本报告分析了传统方法不能够保持球对称性的原因,设计了能够保持 球对称性的离散纵标格式,并通过数值算例进行了验证了。 ********************************************************* Junxia Wei Institute of Applied Physics and Computational Mathematics, No.2, Fenhao Donglu, Haidian District, Beijng , China Title: A Discontinuous Galerkin Method For Neutron Transport Equations on 3-D Unstructured Grids (joint by Wei Junxia ,Yang Shulin, Cheng Jie, Hong Zhenying and Chen Yibing) Abstract: Time-dependent neutron transport equation is a kind of important hyperbolic partial differential equation in nuclear science and engineering applications. High dimension neutron transport calculation include computing of space grid, angle direction, energy group and time step, is very complex and huge scale scientific calculation problem. Discontinuous finite element discrete ordinates (DFE-Sn) method is very efficient for solution of such equations especially while concerned with complicated physics including multimedia, larger grid distortion, complex initial and boundary conditions. In this paper, the discrete scheme of Sn discrete ordinate and discontinuous finite method 3-D unstructured tetrahedral meshes are presented. we developed a serial solver with DFE-Sn method to solve time-dependent neutron transport equations on unstructured tetrahedral grids. Domain decomposition scheme and parallel Sn sweep algorithm on unstructured grids are adopted to improve the efficiency , the parallel computation for the scheme is realized on MPI systems. Numerical experiments demonstrate the accuracy and efficiency of these methods. Key words: discontinuous finite element method, transport equations, unstructured meshes, domain decomposition, sweep algorithm ********************************************************* Wei Xiang City University of Hong Kong Title: Weakly nonlinear geometric optics for hyperbolic systems of conservation laws Abstract: We present a new approach to analyze to validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolilc systems of conservation laws whose eigenvalues are allowed to have constant multiplicity and corresponding characteristic fields to be linearly degenerate. The approach is based on our careful construction of more accurate auxiliary approximation to weakly nonlinear geometric optics, the properties of wave front tracking approximate solutions, the behavior of solutions to the approximate asymptotic equations, and the standard semigroup estimate. This implies that the simpler geometric optics expansion functions can be employed to study the behavior of general entropy solutions to hyperbolic systems of conservation laws. ********************************************************* Kun Xu Hong Kong University of Science and Technology Title: Recent developments of unified gas-kinetic schemes Abstract: In this talk, we are going to introduce recent developments of unified gas-kinetic schemes (UGKS). The new progress of UGKS includes finite difference cut-cell method for flow simulation around complex geometry, multicomponent flows, and multi-scale method for plasma simulations. ********************************************************* Tong Yang City University of Hong Kong Title: Some Recent Studies on the Boltzmann Equation without Angular Cutoff Abstract: After reviewing some recent progress on the Boltzmann equation without angular cutoff, we will present some recent progress on the measure valued solutions. The celebrated H-Theorem implies that large time behavior of solutions to the Boltzmann equation is given by an equilibrium state given in the form of a Maxwellian distribution. This has been justified in verious settings together with convergence rate when the initial energy is finite. On the other hand, when the initial energy is infinite, other time asymptotic states in the form of self-similar solutions are ob- tained and the convergence is justified for the spatially homogeneous Boltzmann equation with Maxwellian type cross sections for the measure valued solutions. This is a joint work with Yoshinori Morimoto and Huijiang Zhao. ********************************************************* Xiongfeng Yang Shanghai Jiao Tong University Title: The existence of boundary layer solution for the Boltzmann equation. Abstract: In this talk, I will discuss the existence of the boundary layer solution for Boltzmann equation. It could be obtained in the following two steps. Firstly, we look for solutions of a steady Boltzmann equation in half space with assigned incoming distribution. Secondly, we establish the admissible conditions for the fixed Maxwellian at the far field. ********************************************************* Guangwei Yuan Institute of Applied Physics and Computational Mathematics 题目: 辐射流场的自适应计算方法 摘要: 将针对应用领域中辐射流场的精密化数值模拟,简要介绍计算方法的研究现状以 及若干相关的研究进展。 ********************************************************* Huijiang Zhao Wuhan University Title: The Vlasov-Maxwell-Boltzmann system for the whole range of cutoff potentials. Abstract: This talk is concerned with the construction of global solutions near Maxwellians to the Cauchy problem of the Vlasov-Maxwell-Boltzmann system for the whole range of curoff potentials. It is based on some recent results joint with Renjun Duan, Yuanjie Lei, and Tong Yang. *********************************************************