Extrapolation methods for approximating arc length and surface area

A well-known method of estimating the length of a parametric curve in is to sample some points from it and compute the length of the polygon passing through them. In this paper we show that for uniform sampling of regular smooth curves, Richardson extrapolation can be applied repeatedly giving a sequence of derivative-free length estimates of arbitrarily high orders of accuracy. A similar result is derived for the approximation of the area of parametric surfaces.      

The admissible interval for the invariant factors of a product of matrices

It is well known that Littlewood-Richardson sequences give a combinatorial characterization for the invariant factors of a product of two matrices over a principal ideal domain. Given partitions a and c, let LR(a,c) be the set of partitions b for which at least one Littlewood - Richardson sequence of type (a,b,c) exists. I. Zaballa has shown in [20] that LR(a, c) has a minimal element w and a maximal element n, with respect to the order bf majorization, depending only on a and c;. In generalLR(a, c) is not the whole interval [w, n]. Here a combinatorial algorithm is provided for constructing all the elements of LR(a, c). This algorithm consists in starting with the minimal Littlewood-Richardson sequence of shape c/a and successively modifying it until the maximal Littlewood - Richardson sequence of shape c/a is achieved. Also explicit bijections between Littlewood - Richardson sequences of conjugate shape and weight and between Littlewood - Richardson sequences of dual shape and equal weight are presented. The bijections are denned by means of permutations of Littlewood - Richardson sequences.     

临界理查森数对不稳定状态下边界层模拟的影响

在边界层参数化方案中,临界理查森数可以用来判断湍流状态和计算边界层高度。大部分模式会将临界理查森数设置为常数,但实际上它会发生改变。本文基于WRF（Weather Research and Forecasting）中尺度数值预报模式和YSU（Yonsei University）边界层参数化方案（以下简称WRF/YSU）,以一次夏季的晴好天气过程为背景,研究了当临界理查森数发生变化,会对气象场和边界层湍流产生的影响。结果表明：（1）当临界理查森数增加时,大范围区域的边界层的高度会增加。在时空上能够体现出来。在空间上,边界层高度分布不均。从时间上看,在下午边界层高度受到的影响最大;（2）理查森数的临界值增加,湍流扩散系数也增大,水汽混合比增加,并且地面热通量总体增大,又会进一步使边界层高度增大;（3）YSU方案中湍流扩散方程由一个局部项、一个非局部项和一个夹卷项组成。对于热通量,夹卷项对总热通量的贡献与局部项和非局部项的贡献相当。对于水汽和动量通量,局地项及夹卷通量项在边界层中最大。     

Richardson-extrapolated sequential splitting and its application

During numerical time integration, the accuracy of the numerical solution obtained with a given step size often proves unsatisfactory. In this case one usually reduces the step size and repeats the computation, while the results obtained for the coarser grid are not used. However, we can also combine the two solutions and obtain a better result. This idea is based on the Richardson extrapolation, a general technique for increasing the order of an approximation method. This technique also allows us to estimate the absolute error of the underlying method. In this paper we apply Richardson extrapolation to the sequential splitting, and investigate the performance of the resulting scheme on several test examples.     

The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition

The Dirichlet problem for a singularly perturbed parabolic reaction-diffusion equation with a piecewise continuous initial condition in a rectangular domain is considered. The higher order derivative in the equation is multiplied by a parameter ?², where ? ∈ (0, 1]. When ? is small, a boundary and an interior layer (with the characteristic width ?) appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the reduced equation passing through the discontinuity point of the initial function; for fixed ?, these layers have limited smoothness. Using the method of additive splitting of singularities (induced by the discontinuities of the initial function and its low-order derivatives) and the condensing grid method (piecewise uniform grids that condense in a neighborhood of the boundary layers), a finite difference scheme is constructed that converges ?-uniformly at a rate of O(N ^?2ln² N + n ₀ ^?1 ), where N + 1 and N ₀ + 1 are the numbers of the mesh points in x and t, respectively. Based on the Richardson technique, a scheme that converges ?-uniformly at a rate of O(N ^?3 + N ₀ ^?2 ) is constructed. It is proved that the Richardson technique cannot construct a scheme that converges in ?-uniformly in x with an order greater than three.     

k-kl模型在风生混合流中的应用

为了考察k-kl模型在风生混合流的应用效果,将影响其紊动扩散模拟结果的因素归结为3类:一是稳定函数的不同;二是壁函数的不同;三是稳态里查德森数Rist的不同.从这3个方面着手对k-kl模型进行分析,比较各种不同方法对混合层及紊动扩散的影响,并采用国外试验结果来检验数值模拟结果.结果表明:稳定函数的选取对计算结果会有影响,相比之下,较新的模型如CA,CB,CH能得出比传统的MY,KC更精确的结果;抛物改进型的壁函数的结果稍好于抛物线型和对称型壁函数;稳态里查德森数对混合层的发展有重要影响,当它的值增大时,同时刻混合层深度也会增加.     

N阶导数值计算的外推算法研究

研究了函数N阶导数的渐进展开,在此基础上提出了基于里查森外推计算方法,并实现了N阶导数的数值计算,实验结果表明提出的算法具有有效性.     

自省与他审：《帕梅拉》的女德建构

塞缪尔·理查逊的道德小说《帕梅拉》中塑造了具有坚定贞洁信仰的女性帕梅拉。他通过意识差异、融合而引起的矛盾对话反映帕梅拉的女性美德自省,并利用大众阅读后产生的心理共鸣进行道德说教。具有女性意识的帕梅拉的自我塑造及自我提升过程反映了的中产阶级上升论及关怀女性的道德论。     

Numerical study on mixed convection from a constant wall temperature circular cylinder in zero-mean velocity oscillating cooling flows

Fluid flow and heat transfer of mixed convection from a constant wall temperature circular cylinder in zero-mean velocity oscillating cooling flows have been simulated based on the projection method with two dimensional exponential stretched staggered cylindrical meshes. Cycle mean temperature and secondary streaming are obtained by the method of partial sums of the Fourier series. Present numerical results are validated by comparing the heat transfer results of free convection and the secondary streaming of pure oscillating flow over a circular cylinder to published experimental and numerical results. The complete structures of the cycle mean temperature and secondary streaming patterns are provided by numerical simulations over wide ranges of the Reynolds number, the Keulegan–Carpenter number and the Richardson number. Based on turning points of the curves of the overall Nusselt numbers versus Reynolds numbers and the characteristics of the cycle averaged temperature and flow patterns, the heat transfer can be divided into three linear regimes (conduction, laminar convection, and turbulent convection dominated regimes) and two non-linear transition regimes. The effects of wave directions, amplitudes, frequencies, and buoyancy forces on the enhancement of heat transfer are also investigated. The effective ranges of the governing parameters for heat transfer enhancement are identified.     

基于MST雷达垂直风速的大气温度剖面反演

本文主要利用MST(mesosphere-stratosphere-troposphere)雷达垂直风速时间序列进行频谱分析,计算B-V(Brunt-V?is?l?)频率,再根据B-V频率和温度的关系,本文建立了离散的温度反演模型,最后反演得到了全高度的温度剖面.通过计算温度剖面和探空仪实测温度剖面比较可见,二者的符合度非常高,变化趋势也完全一致.本文还进一步讨论了,利用B-V频率和MST雷达的水平风速还可以计算理查德森数,通过理查德森数可以很清晰的判断大气的稳定性,进而可以量化大气很多动力学特征以及解释大气中的很多波动现象.因此,B-V频率的获得是MST雷达对大气动力学研究的又一贡献,它可以准确地反演大气的温度剖面,同时获取大气动力学稳定性参数.