关于高阶精度WCNS格式的无粘通量分裂方法（英文）

高阶精度加权紧致非线性格式(WCNS)越来越广泛地应用于复杂流动数值模拟.WCNS可以与多种无粘通量分裂方法结合起来使用.但是,常见的通量分裂方法都是基于低阶格式发展起来的,目前还不清楚哪些通量分裂方法最适合WCNS,也不知道这些方法与高阶格式结合时将会产生什么效果.表面热流计算是高超声速流动数值模拟的难点之一,为了在热流计算时选择合适的通量,研究了多种通量分裂方法的耗散大小.每种通量都可以表示成中心部分与耗散部分之和.这些通量的中心部分相同且非常简单,但是耗散部分较为复杂,且不同的通量分裂方法可导致不同的耗散表达式.通过对通量耗散进行分析可以发现耗散大小与网格界面两侧的物理量跳跃近似线性正相关.数值计算表明高阶格式得到的网格界面左右两侧的物理量跳跃通常远比低阶格式小,因而带来的通量耗散小.通过3个典型算例考察了通量耗散对热流计算的影响,其中包括高超激波/边界层干扰算例.基于对van Leer通量、Steger-Warming通量、KFVS通量、Roe通量、AUSM类通量和HLL类通量的考察,给出了通量选择建议.     

基于AUSM分裂的二维通量分裂格式

基于对流迎风分裂思想构造的AUSM类格式具有简单、高效、分辨率高等优点,在计算流体力学中得到了广泛的应用.传统的AUSM类格式在计算界面数值通量时只考虑网格界面法向的波系,忽略了网格界面横向波系的影响.使用Liou-Steffen通量分裂方法将二维Euler方程的通量分裂成对流通量和压力通量,采用AUSM格式来分别计算对流数值通量和压力数值通量.通过求解考虑了横向波系影响的角点数值通量来构造一种真正二维的AUSM通量分裂格式.在计算一维算例时,该格式保留了精确捕捉激波和接触间断的优点.在计算二维算例时,该格式不仅具有更高的分辨率而且表现出更好的鲁棒性,可以消除强激波波后的不稳定现象.此外,在多维问题的数值模拟中,该格式大大地提高了稳定性CFL数,具有更高的计算效率.因此,它是一种精确、高效并且强鲁棒性的数值方法.     

一种基于TV分裂的真正多维Riemann解法器

给出了一种真正多维的HLL Riemann解法器.采用TV(Toro-Vázquez)分裂将通量分裂成对流通量和压力通量,其中对流通量的计算采用类似于AUSM格式的迎风方法,压力通量的计算采用波速基于压力系统特征值的HLL格式,并将HLL格式耗散项中的密度差用压力差代替,来克服传统的HLL格式不能分辨接触间断的缺点.为了实现数值格式真正多维的特性,分别计算网格界面中点和角点上的数值通量,并且采用Simpson公式加权中点和角点上的数值通量来得到网格界面上的数值通量.采用基于SDWLS(solution dependent weighted least squares)梯度的线性重构来获得空间的二阶精度,时间离散采用二阶Runge-Kutta格式.数值实验表明,相比于传统的一维HLL格式,该文的真正多维HLL格式具有能够分辨接触间断,消除慢行激波波后振荡以及更大的时间步长等优点.并且,与其他能够分辨接触间断的格式（例如HLLC格式）不同的是,真正多维的HLL格式在计算二维问题时不会出现数值激波不稳定现象.     

一种健壮的低耗散通量分裂格式

随着计算流体力学的快速发展,设计精确、高效并且健壮的数值格式变得尤为重要.通过对3种流行的通量分裂方法(AUSM、Zha-Bilgen和Toro-Vázquez)的对流通量和压力通量进行特征分析,构造了一种简单、低耗散并且健壮的通量分裂格式(命名为R-ZB格式).采用Zha-Bilgen分裂方法将Euler方程的通量分裂成对流通量和压力通量,其中对流通量采用迎风方法来计算,压力通量采用低耗散的HLL格式来计算,从而克服了原始的HLL格式不能精确分辨接触间断的缺点.数值实验表明,该文给出的R-ZB格式不仅保留了原始Zha-Bilgen格式简单高效、能够精确分辨接触间断等优点,而且具有更好的健壮性,在计算二维问题时不会出现数值激波不稳定现象.     

Lagrange非结构网格高阶交错型守恒气体动力学格式

提出Lagrange(拉格朗日)非结构网格高阶交错型守恒气体动力学格式.用产生于当前时刻子网格密度和网格声速的子网格压力和MUSCL方法构造了高阶子网格力,利用高阶子网格力构造了高阶空间通量,借助时间中点通量的Taylor(泰勒)展开完成了高阶时间通量离散.研制了Lagrange非结构网格高阶交错型守恒气体动力学格式.对Saltzman活塞问题等进行了数值模拟,数值结果显示了Lagrange非结构网格高阶交错型守恒气体动力学格式的有效性和精确性.     

二维系数不连续Helmholtz方程的高阶紧致差分格式

针对二维系数不连续Helmholtz方程,提出和研究了高阶紧致差分格式,在波数跳跃位置引入局部网格加密技巧进行网格加密.数值实验验证,该高阶紧致差分格式用于求解二维系数不连续Helmholtz方程可以达到四阶精度,局部网格加密技巧能够有效地提高数值解的精度.     

多项式基函数法

提出一种新型的数值计算方法--基函数法.此方法直接在非结构网格上离散微分算子,采用基函数展开逼近真实函数,构造出了导数的中心格式和迎风格式,取二阶多项式为基函数,并采用通量分裂法及中心格式和迎风格式相结合的技术以消除激波附近的非物理波动,构造出数值求解无粘可压缩流动二阶多项式的基函数格式,通过多个二维无粘超音速和跨音速可压缩流动典型算例的数值计算表明,该方法是一种高精度的、对激波具有高分辨率的无波动新型数值计算方法,与网格自适应技术相结合可得到十分满意的结果.     

基于高阶空间精度格式和嵌套网格对旋翼尾迹捕捉的改进

发展了一种基于高阶迎风格式和嵌套网格捕捉直升机悬停旋翼涡尾迹的方法．无粘通量采用Roe Reimann求解器,使用改进的5阶加权基本无振荡(WENO)格式对交界面左右状态进行高阶插值,并与MUSCL插值进行比较．为便于捕捉尾迹和实施周期性边界条件,计算采用结构嵌套网格,其中高质量的旋翼网格完全嵌套于背景网格中．当解达到近似收敛后在桨尖涡分布区域对背景网格进行加密,如此经过3次得到优化的背景网格．考虑到WENO格式插值的特点,提出了搜索3层洞边界和人工外边界的方法以便插值的直接进行．用该方法对一跨音速和一亚音速悬停旋翼粘性流场进行了数值计算．数值结果表明:所发展方法对涡尾迹具有很高的捕捉能力;与MUSCL格式相比,WENO格式具有较低的数值耗散．     

Lagrange柱坐标高阶中心型守恒格式

提出Lagrange柱坐标高阶中心型守恒格式.基于用对守恒律的单调迎风算法(MUSCL)构造的高阶子网格压力,引入了柱坐标高阶体权子网格力和柱坐标高阶面权子网格力,构造了柱坐标高阶体权中心型守恒格式和柱坐标高阶面权中心型格式.柱坐标高阶体权中心型守恒格式满足动量守恒、能量守恒,但不能确定保持一维球对称性.柱坐标高阶面权中心型格式满足能量守恒,保持一维球对称性.两种格式里,格点速度以与网格面的数值通量相容的方式计算.对Saltzman活塞问题等进行了数值模拟,数值结果显示Lagrange柱坐标高阶中心型守恒格式的有效性和精确性.     

一维多介质可压缩流动的守恒型界面追踪方法

本文针对一维问题的ProntTracking方法,提出了一种较易实现的守恒型界面追踪方法.利用双波近似求解Riemann问题来确定界面处的数值通量,在固定的网格上采用统一的有限体积格式进行内点和交界面点的计算,通过守恒插值以及守恒量的重新分配,保证数值解在全场实现一致守恒,将该方法应用于一维多介质可压缩流动的模拟,给出了满意的数值模拟结果.     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

CONTENTS OF VOLUME 30

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Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].