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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.
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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.
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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.
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Ning Jiang
Tsinghua University
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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)
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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。
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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.
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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.
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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.
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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。
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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
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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.
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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.
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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.
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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.
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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。
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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.
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