CONVERGENCE OF AN IMMERSED INTERFACE UPWIND SCHEME FOR LINEAR ADVECTION EQUATIONS WITH PIECEWISE CONSTANT COEFFICIENTS Ⅱ: SOME RELATED BINOMIAL COEFFICIENT INEQUALITIES

In this paper we give proof of three binomial coefficient inequalities. These inequalities are key ingredients in [Wen and Jin, J. Comput. Math. 26, （2008）, 1-22] to establish the L^1-error estimates for the upwind difference scheme to the linear advection equations with a piecewise constant wave speed and a general interface condition, which were further used to establish the L^1-error estimates for a Hamiltonian-preserving scheme developed in [Jin and Wen, Commun. Math. Sci. 3, （2005）, 285-315] to the Liouville equation with piecewise constant potentials [Wen and Jin, SIAM J. Numer. Anal. 46, （2008）, 2688-2714].     

复局部β-凸空间l~β与L~β[0,1]的共轭锥的次表示定理

本文研究复局部β-凸空间l~β与L~β[0,1](0<β<1)的共轭锥(l~β)_β~*与(L~β[0,1])_β~*的构造与表示问题,得到(l~β)_β~*(?)mM_β~+(T),(L~β[0,1])_β~*(?)L~∞M_β~+(I×T),称为(l~β)_β~*与(L~β[0,1])_β~*的次表示定理。     

非交换Lipschitz-φ算子代数

本文引入由紧距离空间(K,d)到给定Banach代数A中的Lipschitz-φ算子构成的非交换Banach代数L~φ(K,A)与l~φ(K,A),证明了它们都是由K到A的全体连续算子构成的非交换Banach代数C(K,A)的子代数,并且关于范数||f||φ=L_φ(f)+||f||∞是Banach代数,研究了不同 Lipschitz尺度函数φ对应的大(小)Lipschitz代数之间的关系。特别当φ(t)=t~α时,引入了极限代数lim_(α→0+)l~α(K,A),lim_(α→+∞)l~α(K,A),lim_(α→0+)L~α(K,A)与lim_(α→+∞)L~α(K,A)以及距离空间的Lipschitz连通性,得到了lim_(α→+∞)l~α(K,A)=A的充要条件,也给出了lim_(α→0+)L~α(K,A)=C(K,A)的条件。     

关于“追赶”法的稳定性

序言 用差分方程逼近常微分方程边值问题,或用隐式差分格式逼近演化型偏微分方程初边值问题时,通常需求解差分方程的两点边值问题.常用的方法是“追赶法”.在[1—4]中,讨论了各种类型的“追赶”法及其稳定性.在这些文章中,或依据系数矩阵特征值的性质,或依据差分方程两点边值问题在C模意义下的性态,来证明“追赶”法的稳定性.关于差分     

Asymptotic Preserving Schemes for Semiconductor Boltzmann Equation in the Diffusive Regime

As is known, the numerical stiffness arising from the small mean free path is one of the main difficulties in the kinetic equations. In this paper, we derive both the split and the unsplit schemes for the linear semiconductor Boltzmann equation with a diffusive scaling. In the two schemes, the anisotropic collision operator is realized by the "BGK"-penalty method, which is proposed by Filbet and Jin [F. Filbet and S. Jin, J. Comp. Phys. 229(20), 7625-7648, 2010] for the kinetic equations and the related problems having stiff sources. According to the numerical results, both of the schemes are shown to be uniformly convergent and asymptotic-preserving. Besides, numerical evidences suggest that the unsplit scheme has a better numerical stability than the split scheme.     

THE ASYMPTOTIC PRESERVING UNIFIED GAS KINETIC SCHEME FOR GRAY RADIATIVE TRANSFER EQUATIONS ON DISTORTED QUADRILATERAL MESHES

In this paper, we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes. Different from the former scheme [J. Comput. Phys. 285(2015), 265-279] on uniform meshes, in this paper, in order to obtain the boundary fluxes based on the framework of unified gas kinetic scheme (UGKS), we use the real multi-dimensional reconstruction for the initial data and the macro-terms in the equation of the gray transfer equations. We can prove that the scheme is asymptotic preserving, and especially for the distorted quadrilateral meshes, a nine-point scheme [SIAM J. SCI. COMPUT. 30(2008), 1341-1361] for the diffusion limit equations is obtained, which is naturally reduced to standard five-point scheme for the orthogonal meshes. The numerical examples on distorted meshes are included to validate the current approach.     

An exact solution procedure for a cluster hub location problem

This paper proposes a new (MIP) model formulation and a new solution procedure for the hub network design problem under a non-restrictive policy introduced by Sung and Jin [Sung, C.S., Jin, H.W., 2001. Dual-based approach for a hub network design problem under non-restrictive policy. European Journal of Operational Research 132 (1), 88–105]. The model formulation contains significantly fewer variables so that optimal solutions for the LP-relaxation of the model can be determined for large instances using standard procedures for LP-models. Furthermore, the LP-relaxation provides very tight lower bounds. Computational results are given, which demonstrate that the new model formulation allows for solving much larger instances. It turned out that the new (exact) solution procedure, which utilises the new model formulation, is faster than the heuristic proposed by Sung and Jin (2001). It is also shown that the problem is np-hard.     

赋范空间E和l~1(Γ)的单位球面间等距映射的延拓

研究赋范空间E和l~1(Γ)的单位球面之间的等距映射的延拓,得到E和l~1(Γ)的单位球面之间的满等距映射可以延拓为全空间E上的实线性等距算子,从而肯定地回答了相应的Tingley问题.     

A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space

Considering the features of the fractional Klein-Kramers equation (FKKE) in phase space, only the unilateral boundary condition in position direction is needed, which is different from the bilateral boundary conditions in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. In the paper, a finite difference scheme is constructed, where temporal fractional derivatives are approximated using L1 discretization. The advantages of the scheme are: for every temporal level it can be dealt with from one side to the other one in position direction, and for any fixed position only a tri-diagonal system of linear algebraic equations needs to be solved. The computational amount reduces compared with the ADI scheme in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and the five-point scheme in [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. The stability and convergence are proved and two examples are included to show the accuracy and effectiveness of the method.     

A Relaxation Scheme for Solving the Boltzmann Equation Based on the Chapman-Enskog Expansion

Abstract In [16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jinand Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weaklyparabolic, has a linearly hyperbolic convection part, and is endowed with a generalized eotropy inequality. Itagrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion. We develop a one-dimensional non-oscillatory numerical scheme based on the relaxed Burnett system forthe Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme,and those obtained by the DSMC (Direct Simulation Monte Carlo), by the Navier-Stokes equations and bythe extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments showthat the relaxed Burnett gives more accurate approximations to the shock profiles of the Boltzmann equationobtained by the DSMC, for a range of Mach numbers for hypersonic flows, th     

本刊英文版Vol.30(2014),No.5论文摘要（英文）

<正>Similarity Degrees for the Crossed Product of yon Neumann Algebras Jin Song WU Wen Ming WU Abstract In this paper,we will estimate an upper bound for the similarity degree of the crossed product of a hyperfinite finite yon Neumann algebra by weakly compact action of an infinite discrete group.We will also improve some upper bounds for similarity degrees of some finite von Neumann algebras.     

THE CAPACITY DENSITY AND THE HAUSDORFF DIMENSION OF FRACTAL SETS

ＴＨＥＣＡＰＡＣＩＴＹＤＥＮＳＩＴＹＡＮＤＴＨＥＨＡＵＳＤＯＲＦＦＤＩＭＥＮＳＩＯＮＯＦＦＲＡＣＴＡＬＳＥＴＳ￥ＸｕＹｏｕ（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ，ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ，Ｓｈａｎｇｈａｉ２００４３３，Ｓｈａｎｇｈａｉ，...     

本刊英文版Vol.31(2015),No.4论文摘要（英文）

<正>Nonorientable Genera of Petersen Powers Wen Zhong LIU Ting Ru SHEN Yi Chao CHEN Abstract In the paper,we prove that for every integern≥1,there exists a Petersen power P~n with nonorientable genus and Euler genus precisely n.which improves the upper bound of Mohar and Vodopivec's result[J.Graph Theory,67,1-8(2011)]that for every integer k(2≤k≤n-1),a Petersen power P~n exists with nonorientable genus and Euler genus precisely k.Polynomials with Palindromic and Unimodal Coefficients     

矩阵方程X+A~*X~(-q)A=I(q>0)的Hermite正定解

1.引言 本文研究矩阵方程 X+A*X-qA=I (1)的Hermite正定解,其中I是一个n×n阶单位矩阵, A是一个n×n阶复矩阵, q是实数且q>0.q=1,q=2时的方程是从动态规划,随机过滤,控制理论和统计学中推导出来的,最近已有许多人对此进行了研究(见参考文献[1,2,4]),本文我们将研究方程(1)的解的存在性和解的性质,并讨论迭代求解及迭代解的收敛性. 对于Hermite矩阵X和Y,文中X≥Y表示X-Y是半正定的,X>y表示X-Y是正定的;对于方阵M,M*表示M的共轭转置,ρ(M)表示M的谱半径,λi(M)     

On the $L_\infty$ Convergence and the Extrapolation Method of a Difference Scheme for Nonlocal Parabolic Equation with Natural Boundary Conditions

In paper [4] (J. Comput. Appl. Math.,76 (1996), 137-146), a difference scheme for a class of nonlocal parabolic equations with natural boundary conditions was derived by the method of reduction of order and the unique solvability and second order convergence in $L_2$-norm are proved. In this paper, we prove that the scheme is second order convergent in $L_\infty$ norm and then obtain fourth order accuracy approximation in $L_\infty$ norm by extrapolation method. At last, one numerical example is presented.     

A ROBUST DISCRETIZATION OF THE REISSNER-MINDLIN PLATE WITH ARBITRARY POLYNOMIAL DEGREE

A numerical scheme for the Reissner-Mindlin plate model is proposed.The method is based on a discrete Helmholtz decomposition and can be viewed as a generalization of the nonconforming finite element scheme of Arnold and Falk[SIAM J.Numer.Anal.,26(6):1276-1290,1989].The two unknowns in the discrete formulation are the in-plane rotations and the gradient of the vertical displacement.The decomposition of the discrete shear variable leads to equivalence with the usual Stokes system with penalty term plus two Poisson equations and the proposed method is equivalent to a stabilized discretization of the Stokes system that generalizes the Mini element.The method is proved to satisfy a best-approximation result which is robust with respect to the thickness parameter t.     

A BGK‐penalization‐based asymptotic‐preserving scheme for the multispecies Boltzmann equation

An asymptotic‐preserving (AP) scheme is efficient in solving multiscale problems where kinetic and hydrodynamic regimes coexist. In this article, we extend the BGK‐penalization‐based AP scheme, originally introduced by Filbet and Jin for the single species Boltzmann equation (Filbet and Jin, J Comput Phys 229 (2010) 7625–7648), to its multispecies counterpart. For the multispecies Boltzmann equation, the new difficulties arise due to: (1) the breaking down of the conservation laws for each species and (2) different convergence rates to equilibria for different species in disparate masses systems. To resolve these issues, we find a suitable penalty function—the local Maxwellian that is based on the mean velocity and mean temperature and justify various asymptotic properties of this method. This AP scheme does not contain any nonlinear nonlocal implicit solver, yet it can capture the fluid dynamic limit with time step and mesh size independent of the Knudsen number. Numerical examples demonstrate the correct asymptotic‐behavior of the scheme. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

An asymptotic preserving scheme with the maximum principle for the M1 model on distorded meshes

In this Note, we show that a recent scheme introduced by Buet et al. (2011) [5] for the nonlinear two moments $M_1$ model of linear transport and which captures correctly the diffusion limit on distorded meshes (AP scheme) also possesses the maximum principle. The main idea of the design of this scheme is to rewrite the model as a gas dynamics model and to use an Eulerian scheme, derived from a Lagrange + remap scheme. To obtain the AP property we use the multidimensional extension, developed by Buet et al. (2012) [6], of the Jin and Levermore (1996) procedure [9] for the hyperbolic heat equation. We will show that this scheme is entropic which ensures the maximum principle of the $M_1$ model. More we present some numerical results, on distorted quadrangular and triangular meshes which show that the scheme is second order in the diffusive regime.     

Fast Evaluation of Linear Combinations of Caputo Fractional Derivatives and Its Applications to Multi-Term Time-Fractional Sub-Diffusion Equations

In the present work, linear combinations of Caputo fractional derivatives are fast evaluated based on the efficient sum-of-exponentials (SOE) approximation for kernels in Caputo fractional derivatives with an absolute error $\epsilon,$ which is a further work of the existing results in [13] (Commun. Comput. Phys., 21 (2017), pp. 650-678) and [16] (Commun. Comput. Phys., 22 (2017), pp. 1028-1048). Both the storage needs and computational amount are significantly reduced compared with the direct algorithm. Applications of the proposed fast algorithm are illustrated by solving a second-order multi-term time-fractional sub-diffusion problem. The unconditional stability and convergence of the fast difference scheme are proved. The CPU time is largely reduced while the accuracy is kept, especially for the cases of large temporal level, which is displayed by numerical experiments.     

A CHARACTERIZATION OF THE EMBEDDABILITY OF GRAPHS ON THE SURFACE OF GIVEN GENUS

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