workshop and summer school on kinetic theory and gas dynamics

Workshop and Summer School on Kinetic Theory and Gas Dynamics 1

Table of Contents

Organization…………………………………………………………………………………………………………………………..2

Objectives……………………………………………………………………………………………………………………………...2

Sponsors………………………………………………………………………………………………………………………...........2

Useful Information……………………………………………………………………………………………………………...2

Venue…………………………………………………………………………………………………………………………………….2

Registration…………….……………………………………………………………………………………………………………….2

Hotel……………………………………………………………………………………………………………………………………….2

Transportation Directions……………………………………………………………………………………………………….3

Internet Access…………………………………………………………………………………………………………………..4

Useful Links……………………………………………………………………………………………………………………………...4

Emergency Contact………………………………………………………………………………………………………………….4

Schedule by Day………………….……………….……..…………………………………………..………………………….5

Abstracts………………………………………………………………………………………………………………………………7


Objectives

The purpose of Workshop and Summer School on Kinetic Theory and Gas Dynamics is to

bring together mathematicians from all over the world in the area of kinetic theory and gas

dynamics to present their recent research results, provide lectures and mini-courses to

Ph.D. students and postdoctoral fellows from top universities in China. The participants

will exchange new ideas and to discuss current challenging issues. The main topics

include:

Theory and computational methods for kinetic equations and nonlinear hyperbolic

systems; applications of kinetic theory; uncertainty quantification.

Workshop Homepage

http://math.sjtu.edu.cn/conference/wssktgd/SJTUWorkshopSchedule.pdf

Sponsors

This is an event of the newly founded Zhiyuan Center of Mathematical Sciences, which is

funded by National Natural Science Foundation of China and Shanghai Jiao Tong

University.

Useful Information

Venue

August 4-7

Lecture Hall (100-L), 1/F, Math Building, SJTU

Address: No. 800, Dongchuan Road, Minhang District, Shanghai, China

(上海交通大学闵行校区地址：上海市闵行区东川路 800 号)

Registration

The registration will be held from 14:00 to 18:00, August 3，Academic Exchange Center

lobby(学术活动中心大厅 ), Shanghai Jiao Tong University. We will also set tables

everyday at the lobby of Math Building, SJTU during the workshop.

Hotel

Academic Exchange Center (学术活动中心)

Minhang Campus, SJTU

800 Dongchuan Rd. Minhang District, Shanghai 200240, China


Tel: 86-21-54740800

Check-in

We have reserved a hotel room for you in your name so that you can directly check in at

the reception once you arrive.

Meals

◎ Breakfast: Hotel rate has already included breakfast. The breakfast will be available

from 7:00am-9:30am.

◎ Lunch: 1/F, Academic Exchange Center (August 4, 5, 6, 7)

◎ Dinner: Liu Yuan (August 4, 5, 6, 7)

Transportation Directions

From Pudong International Airport (PVG) to the Academic Exchange Center, SJTU

Please give the following message to the taxi driver if you don’t speak Chinese:

请送我到闵行区东川路800号上海交通大学学术活动中心。

请走S32（从1号航站楼上S32）到剑川路下，到红绿灯处往右拐上剑川路，左拐至沧源路，

再左拐至东川路，进东川路 800 号（近永平路）大门进，进门后往右行驶50米即可见学术

活动中心。

Please take me to Academic Center of Shanghai Jiao Tong University, located at 800

Dongchuan Road Minhang District

Please take highway S32 (from Terminal 1), exit on Jianchuan Road. Right turn at the

traffic lights to Jianchuan Road, then turn left to Cangyuan Road, followed by left turn to

Dongchuan Road. 800 Dongchuan Road(the gate of SJTU at the cross of Dongchuan

Road and Yongping Road)is on your left, Enter the gate, turn right and the Academic

Center is about 50 meters on your right.

From Hongqiao Airport (SHA) to the Academic Exchange Center, SJTU


请送我到闵行区东川路800号上海交通大学学术活动中心。

请走S4高架到剑川路下，到红绿灯处往右拐上剑川路，左拐至沧源路，再左拐至东川路，

进东川路 800 号（近永平路）大门进，进门后往右，学术活动中心在50米后右手边。

Please take me to Academic Center of Shanghai Jiao Tong University, located at 800

Dongchuan Road Minhang District

Please take highway S4, exit on Jianchuan Road. Right turn at the traffic lights to

Jianchuan Road, then turn left to Cangyuan Road, followed by left turn to Dongchuan

Road. 800 Dongchuan Road is on your left which is the gate of SJTU. Enter the gate, turn

right and the Academic Center is about 50 meters on your right.


From Pudong International Airport (PVG) to the Department of Mathematics, SJTU


请送我到闵行区东川路800号上海交通大学数学楼。

请走S32（从1号航站楼上S32）到剑川路下，到红绿灯处往右拐上剑川路，左拐至沧源路，

再左拐至东川路，进东川路800 号（近永平路）大门进，进门后向左，然后向北直行200

米即可见数学楼。

Please take me to the Department of Mathematics, Shanghai Jiao Tong University,

located at 800 Dongchuan Road Minhang District

Please take highway S32 (from Terminal 1), exit on Jianchuan Road. Right turn at the

traffic lights to Jianchuan Road, then turn left to Cangyuan Road, followed by left turn to

Dongchuan Road. 800 Dongchuan Road (the gate of SJTU at the cross of Dongchuan

Road and Yongping Road) is on your left, turn left and go straight toward north and the

Department of Mathematics is about 200 meters on your right.

From Hongqiao International Airport (SHA) to the Department of Mathematics, SJTU


请送我到闵行区东川路800号上海交通大学数学楼。

请走S4到剑川路下，到红绿灯处往右拐上剑川路，沿剑川路向西，左拐至沧源路，再左拐

至东川路，进东川路 800号（近永平路）大门进，进门后向左，然后向北直行200米即可见

数学楼。

Please take me to the Department of Mathematics, Shanghai Jiao Tong University,

located at 800 Dongchuan Road，Minhang District

Please take highway S4, exit on Jianchuan Road. Right turn at the traffic lights to

Jianchuan Road, then turn left to Cangyuan Road, followed by left turn to Dongchuan

Road. 800 Dongchuan Road (the gate of SJTU at the cross of Dongchuan Road and

Yongping Road) is on your left, turn left and go straight toward north and the Department

of Mathematics is about 200 meters on your right.

Please ask the driver call 13918684080 (cell of Limin Qin) for emergency.

Chinese:司机，如有问题请打秦丽敏手机：13918684080

Internet Access

Wireless internet access is available in the Lecture Hall (100-M).

The user name is mathconference and the password is:12345678

Useful Links

Shanghai Jiao Tong University: www.sjtu.edu.cn

Department of Mathematics, SJTU: www.math.sjtu.edu.cn

Emergency Contact

Contact: Limin Qin (Cell: 13918684080)

http://www.sjtu.edu.cn/

http://www.math.sjtu.edu.cn/


Daily Schedule

Conference Venue: Lecture Hall (100-L), First Floor of the Math Building

Aug.4 (Tue.)

09:50-10:00 Opening

10:00-10:45 Tong Yang, City University of Hong Kong

Measure valued solutions to the Boltzmann equation

10:45-11:15 Coffee break

11:15-12:00 Qin Li, California Institute of Technology

Numerical methods for linear half-space kinetic equations

12:00-14:00 Lunch break

14:00-16:00 David Levermore, University of Maryland

Mini-course lectures 1&2 on "Introduction to Kinetic Theory"

Aug.5 (Wed.)


TBA

09:15-10:00 Doron Levy, University of Maryland

Modeling Selective Local Interactions with Memory


10:30-11:15 Anton Arnold, Technical University of Vienna

Entropy method for hypocoercive & non-symmetric Fokker-Planck

equations with linear drift

11:15-12:00 Jingwei Hu, Purdue University

A stochastic Galerkin scheme for the Boltzmann equation with

uncertainty efficient in the fluid regime


14:00-16:00 Alexander Kurganov, Tulane University

Mini-courses lectures 1&2 on "Introduction to Shock

CapturingMethods"

Aug.6 (Thur.)


Hybrid Finite-Volume-Particle Method for Dusty Gas Flows

09:15-10:00 Alina Chertock, North Carolina State University

A flux-splitting based stochastic Galerkin method for nonlinear

systems of hyperbolic conservation laws with uncertainty


10:30-11:15 Ping Zhang, Chinese Academy of Sciences

Inhomogeneous incompressible viscous flows with slowly varying initial

data

11:15-12:00 Ning Jiang, Tsinghua University

Hydrodynamic models of self-organized dynamics: convergence proof



Mini-course lectures 3&4 on "Introduction to Kinetic Theory"


Aug.7 (Fri.)

08:30-09:15 Peter Markowich, University of Cambridge and KUAST

Three Forward PDE Problems with Urgent Need of Data Assimilation

09:15-10:00 Jose Carrillo, Imperial College, London

Swarming Models with Repulsive-Attractive Effects


10:30-11:15 Weiran Sun, Simon Fraser University

The radiative transfer equation in the forward-peaked regime

11:15-12:00 Xinghui Zhong, Michigan State University

Compact WENO Limiters for Discontinuous Galerkin Methods



Mini-courses lectures 3&4 on "Introduction to Shock Capturing

Methods"


Abstracts


Mini-courses

David Levermore, University of Maryland, College Park

Introduction to Kinetic Theory

1. History and Derivation of the Boltzmann Equation

2. Properties of General Kinetic Equations

3. Fluid Dynamical Regimes

4. Transition Regimes

*********************************************************************

Alexander Kurganov, Tulane University

Robust Finite-Volume Methods for Nonlinear Hyperbolic PDEs

I will first review the main concepts of Godunov-type finite-volume methods for

hyperbolic systems of conservation laws. The key idea behind the construction of this

type of schemes is a global approximation of solutions using piecewise polynomial

functions followed by their time evolution according to the integral form of the

studied hyperbolic system. A success of this approach relies upon a finite speed of

propagation, which allows to consider a local behaviour of the solution only, that is,

the solution can be evolved in time in an upwind manner by solving (generalized)

Riemann problems in localized space-time control volumes.

Constructing piecewise polynomial approximations requires nonlinear limiting

techniques since a straightforward linear approach leads to large O(1) errors near the

cell interfaces, where the difference between the left- and right-sided piecewise

polynomial values is not necessarily proportional to the size of the formal

approximation error. This may happen in the neighbourhoods of discontinuities,

which are typically developed by solutions of nonlinear hyperbolic PDEs even when

the initial data are smooth. Nonlinear limiters are designed to prevent large

oscillations by differentiating the computed solutions in the local direction of

smoothness, which may be different form the local direction of propagation (this

means that stability of high-order schemes cannot be guaranteed just by upwinding).

I will review several nonlinear limiting techniques for both second- and higher-order

piecewise polynomial reconstructions.

The second major ingredient of Godunov-type upwind schemes is a (generalized)

Riemann problem solver, which is quite easy to obtain for linear hyperbolic systems.

However, solving (generalized) Riemann problems for general nonlinear hyperbolic

systems is a highly nontrivial task even in the one-dimensional case. Therefore, one

may be interested in designing Riemann-problem-solver-free finite-volume methods,

which then can be used as an efficient and robust black-box solvers for a variety of


hyperbolic problems.

A popular class of Riemann-problem-solver-free methods is Godunov-type central

schemes. They are designed in such a way that solving (generalized) Riemann

problems is avoided by considering control volumes that contain all (possibly

discontinuous) nonlinear waves, which are averaged at the end of each evolution

step. The resulting schemes are robust and can be made very accurate using

high-order piecewise polynomial approximations. However, central schemes contain

more numerical dissipation than their upwind counterparts and thus they often fail

to achieve high resolution of linear contact waves developed by solutions of

nonlinear hyperbolic systems.

Starting 2000, we have been developing a new class of Godunov-type central

schemes, central-upwind schemes. These methods use the same machinery as their

central predecessors, but the control volumes are selected in a more accurate way by

taking into account local speeds of propagation (this brings a taste of upwinding into

this class of central schemes, which are therefore called central-upwind schemes).

This allows to significantly reduce the numerical dissipation without risking

oscillations.

Central-upwind schemes are constructed in three steps: piecewise polynomial

reconstruction, evolution of the reconstructed solution on a new, refined grid and

projection of the evolved solution back onto the original grid. The resulting schemes

are quite cumbersome (though they are still robust and can be used as a black-box

solver for general multidimensional hyperbolic systems), but they can be significantly

simplified by reduction to a semi-discrete form, that is, by taking the limits as the size

of timestep shrinks to zero. I will show how both one- and two-dimensional fully

discrete and semi-discrete schemes are derived. In the two-dimensional case, I will

consider both rectangular and triangular grids.

At the end, I will present central-upwind schemes for hyperbolic systems of balance

laws. As an example, I will consider the Saint-Venant system of shallow water

equations. The key issues in designing robust and efficient central-upwind schemes

will be their ability to preserve a delicate balance between the flux and source terms

(this is extremely important since in many geophysical applications, relevant

solutions are small perturbations of steady states) and positivity of physical

quantities that are supposed to remain positive (nonnegative) for all times (this

feature is also of a great practical importance since one often has to handle dry and

almost dry areas).


Workshop Lectures

Anton Arnold, Technical University of Vienna

Entropy method for hypocoercive & non-symmetric Fokker-Planck equations with

linear drift

In the last 15 years the entropy method has become a powerful tool for analyzing the

large-time behavior of the Cauchy problem for linear and non-linear Fokker-Planck

type equations (advection-diffusion equations, kinetic Fokker-Planck equation of

plasma physics, e.g.). In particular, this entropy method can be used to analyze the

rate of convergence to the equilibrium (in relative entropy and hence in L1). The

essence of the method is to first derive a differential inequality between the first and

second time derivative of the relative entropy, and then between the entropy

dissipation and the entropy.

For degenerate parabolic equations, the entropy dissipation may vanish for states

other than the equilibrium. Hence, the standard entropy method does not carry over.

For hypocoercive Fokker-Planck equations (with drift terms that are linear in the

spatial variable) we introduce an auxiliary functional (of entropy dissipation type) to

prove exponential decay of the solution towards the steady state in relative entropy.

We show that the obtained rate is indeed sharp (both for the logarithmic and

quadratic entropy). Finally, we extend the method to the kinetic Fokker-Planck

equation (with non-quadratic potential) and non-degenerate, non-symmetric

Fokker-Planck equations. For the latter examples the "hypocoercive entropy method"

yields the sharp global decay rate (as an envelope for the relative entropy function),

while the standard entropy method only yields the sharp local decay rate.

References:

1) A. Arnold, J. Erb. Sharp entropy decay for hypocoercive and non-symmetric

Fokker-Planck equations with linear drift, arXiv 2014.

2) A. Arnold, P. Markowich, G. Toscani, A. Unterreiter. On logarithmic Sobolev

inequalities and the rate of convergence to equilibrium for Fokker-Planck type

equations. Comm. PDE 26/1-2 (2001) 43-100.


Jose Carrillo, Imperial College

Swarming Models with Repulsive-Attractive Effects

I will present a survey of the main results about first and second order models of

swarming where repulsion and attraction are modeled through pairwise potentials.

We will mainly focus on the stability of the fascinating patterns that you get by

random data particle simulations, flocks and mills, and their qualitative behavior.

*********************************************************************

Alina Chertock, North Carolina State University

A flux-splitting based stochastic Galerkin method for nonlinear systems of

hyperbolic conservation laws with uncertainty

We introduce a flux-splitting based stochastic Galerkin methods for nonlinear

systems of hyperbolic conservation/balance laws with random inputs. The method

uses a generalized polynomial chaos approximation in the stochastic Galerkin

framework (referred to as the gPC-SG method). It is well-known that such

approximations for nonlinear system of hyperbolic conservation laws do not

necessarily yield globally hyperbolic systems: the Jacobian may contain complex

eigenvalues and thus trigger instabilities and ill-posedness. In this paper, we present

a systematic way to overcome this difficulty. The main idea is to split the underlying

system into a linear hyperbolic system, and a nonlinear degenerated hyperbolic

system which can be solved successively as scalar conservation laws with variable

coefficients and source terms. The gPC-SG method, when applied to each of these

subsystems, result in globally hyperbolic systems. The performance of the new

gPC-SG method is illustrated on a number of numerical examples.

*********************************************************************

Jingwei Hu, Purdue University

A stochastic Galerkin scheme for the Boltzmann equation with uncertainty efficient

in the fluid regime

We develop a stochastic Galerkin method for the nonlinear Boltzmann equation with

uncertainty. The method is based on the generalized polynomial chaos (gPC) and can

handle random inputs from collision kernel, initial data or boundary data. We show

that a simple singular value decomposition of gPC related coefficients combined with

the Fourier-spectral method (in velocity space) allows one to compute the collision

operator efficiently. When the Knudsen number is small, we propose a new

technique to overcome the stiffness. The resulting scheme is uniformly stable in both

kinetic and fluid regimes, which offers a possibility of solving the compressible Euler

equation with random inputs.


Ning Jiang, Tsinghua University

Hydrodynamic models of self-organized dynamics: convergence proof.

In this talk, we will review the hydrodynamic models of self-organized dynamics,

whose formal derivation and some existence were proposed and proved by

Demond-Liu and their collaborators. We justify rigrously this macroscopic limit from

the kinetic self-organized models, employing the Hilbert expansion. It is a joint work

with Linjie Xiong and Tengfei Zhang.

*********************************************************************

Alexander Kurganov, Tulane University

Hybrid Finite-Volume-Particle Method for Dusty Gas Flows

We study the dusty gas flow modeled by the two-phase system composed of a

gaseous carrier (gas phase) and a particulate suspended phase (dust phase). The gas

phase is modeled by the compressible Euler equations of gas dynamics and the dust

phase is modeled by the pressureless gas dynamics equations. These two sets of

conservation laws are coupled through source terms that model momentum and

heat transfers between the phases. When an Eulerian method is adopted for this

model, one can notice the obtained numerical results are typically significantly

affected by numerical diffusion. This phenomenon occurs since the pressureless gas

equations are nonstrictly hyperbolic and have a degenerate structure in which

singular delta shocks are formed, and these strong singularities are vulnerable to the

numerical diffusion.

We introduce a low dissipative hybrid finite-volume-particle method in which the

compressible Euler equations for the gas phase are solved by a central-upwind

scheme, while the pressureless gas dynamics equations for dust phase are solved by

a sticky particle method. The obtained numerical results demonstrate that our hybrid

method provides a sharp resolution even a relatively small number of particle is

used.

*********************************************************************

Doron Levy, University of Maryland

Modeling Selective Local Interactions with Memory

Motivated by phototaxis, we consider a system of particles that simultaneously move

on 1D or 2D periodic lattice at discrete times steps. Particles remember their last

direction of movement and may either choose to continue moving in this direction,

remain stationary, or move toward one of their neighbors. The form of motion is

chosen based on predetermined stationary probabilities. Our results demonstrate a

connection between these probabilities and the emerging patterns and size of

aggregates. We develop a reaction diffusion master equation from which we derive a


system of ODEs describing the dynamics of the particles on the lattice. We show that

the ODEs may replicate the aggregation patterns produced by the stochastic particle

model. This is a joint work with Amanda Galante and Dan Weinberg.

*********************************************************************

Qin Li, California Institute of Technology

Numerical methods for linear half-space kinetic equations

Half-space kinetic equation is the model problem to understand the boundary layer

emerging at the interface in the kinetic-fluid coupling. In this talk, we report our

recent progress on analysis and algorithm development for general linear kinetic

equations with general boundary conditions. Applications include neutron transport

equation with multi-frequencies, linearized 2D Boltzmann equations with Maxwell

boundary conditions etc. The coupled system (heat equation coupling with neutron

transport equation) will also be shown.

*********************************************************************

Peter Markowich, University of Cambridge and KAUST

Three Forward PDE Problems with Urgent Need of Data Assimilation

Abstract: I discuss three very different forward PDE problems which need data

assimilation/inverse approaches to make them potentially useful in practical

applications. The first problem is a reaction-diffusion system for biological

transportation network formation and adaptation, the second is a highly

nonstandard parabolic free boundary problem describing price formation in

economic markets and the third problem is the

incompressible Navier-Stokes-Forchheimer-Brinkmann system for flow in porous

media.

*******************************************************

Weiran Sun, Simon Fraser University

The radiative transfer equation in the forward-peaked regime

Abstract: In this work we study the radiative transfer equation in the forward-peaked

regime in free space. Specically, it is shown that the equation is well-posed by

proving instantaneous regularization of weak solutions for arbitrary initial datum in

L^1. Classical techniques for hypo-elliptic operators such as averaging lemma are

used in the argument. Among the interesting aspect of the proof are the use of the

stereographic projection and the presentation of a rigorous expression for the

scattering operator given in terms of a fractional Laplace-Beltrami operator on the

sphere, or equivalently, a weighted fractional Laplacian analog in the projected plane.

Such representations may be used for accurate numerical simulations of the model.

As a bonus given by the methodology, we show convergence of Henyey-Greenstein


scattering models and vanishing of the solution at time algebraic rate due to

scattering diffusion. This is a joint work with Ricardo Alonso.

*********************************************************************

Tong Yang, City University of Hong Kong

Measure valued solutions to the Boltzmann equation

In this talk, we will present our recent results on the measure valuedsolutions to the

space homogeneous Boltzmann equation. We will start from a new classification of

the characteristic functions in the Fourier space according to the moment constraint,

and then show the existence, regularity and large time behavior of solutions with

finite or infinite energy for the Maxwellian molecule cross section. Finally, we will

present the existence for hard and soft potentials with finite energy.

*********************************************************************

Ping Zhang, Chinese Academy of Sciences

Inhomogeneous incompressible viscous flows with slowly varyinginitial data

The purpose of this paper is to provide a large class of initial data which generates

globalsmooth solution of the 3-D inhomogeneous incompressible Navier-Stokes

system in the whole space~$\R^3$. This class of data is based on functions which

vary slowly in one direction. The idea is that 2-D inhomogeneous Navier-Stokes

system with large data is globally well-posedness and we construct the 3-D

approximate solutions by the 2-D solutions with a parameter. One of the key point

of this study is the investigation of the time decay properties of the solutions to the

2-D inhomogeneous Navier-Stokes system. We obtained the same optimal decay

estimates as the solutions of 2-D homogeneous Navier-Stokes system.

*********************************************************************

Xinghui Zhong, Michgan State University

Compact WENO Limiters for Discontinuous Galerkin Methods

Discontinuous Galerkin (DG) method is a class of finite element methods that has

gained popularity in recent years due to its flexibility for arbitrarily unstructured

meshes, with a compact stencil, and with the ability to easily accommodate arbitrary

h-p adaptivity. However, some challenges still remain in specific application problems.

In this talk, we design compact limiters using weighted essentially non-oscillatory

(WENO) methodology for DG methods solving conservation laws, with the goal of

obtaining a robust and high order limiting procedure to simultaneously achieve

uniform high order accuracy and sharp, non-oscillatory shock transitions. The

mainadvantage of these compact limiters is their simplicity in implementation,

especially on multi-dimensional unstructured meshes.

workshop and summer school on kinetic theory and gas dynamics

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