差分格式的余项效应分析及其应用

本文对已有的差分格的色散关系和群速度效应的Ｆｏｕｒｉｅｒ分析提出了置疑，指出症结所在并予以纠正，并且利用差分格式的Ｍｏｄｉｆｉｅｄ ＰＤＥ思想，提出了一种新的构造性差分格式分析方法－差分格式余项效应分析。这种方法基于差分格式的耗散关系和色散关系的，人有明显的构造性和现实意义。     

守恒格式稳定性分析与耗散守恒格式

本文从守恒格式出发,建立分析稳定性和耗散性的启发性方法和Fourier分析方法,给出了耗散守恒格式的严格定义及三点耗散守恒格式的充要条件。应用本文的方法,重新分析了三点格式,得到如下结论:某些常系数耗散格式,在某些情况下,之所以会得到非物理解或发生非线性不稳定,是由于该格式在这些情况下,已经是零耗散的或是负耗散     

Sobolev方程的特征混合有限元方法

本文针对Sobolev方程提出了一种新型数值模拟方法一特征混合有限元方法．该方法对方程的对流部分采用沿特征线的后退差分格式求解，以保证较小的截断误差限并避免了在流动的锋线前沿数值弥散现象的出现；对流动的扩散部分采用最低次混合元方法求解，以保证格式可同时逼近未知函数及伴随向量函数．由于该方法中检验函数可取分片常数，此格式在某种意义上具有局部守恒性质．通过严格的数值分析，建立了格式对待求函数及伴随向量的最优L2误差分析理论．     

非线性Schroedinger方程的高精度守恒差分格式

本文首先分析线性Schroedinger方程一种高阶差分格式的构造方法，得到方程的耗散项．在此基础上对三次非线性Schroedinger方程，提出了一种精度为O(r^2 h^2)的差分格式，证明了该格式保持了连续方程的两个守恒量，且是收敛的与稳定的．并通过数值例子与已有隐格式进行了比较，结果表明，本文格式在计算量类似的情况下，提高了数值精度．     

空间网格上的三次GC^1插值格式

本文得到了由三次参数多项式构造的ＧＣ＾１插值格式，该插值格式定义在由空间三角一菜和空间四边形构成的空间网格上，并通过该网格的所有网点，同时在每个网点处以事先给定的平面为切平面。     

对称正则长波方程的拟紧致守恒差分逼近

该文对称正则长波方程的初边值问题进行了数值研究，提出了一个两层隐式拟紧差分格式，格式很好地模拟了初值问题的守恒性质．得到了差分解的存在唯一性，并在先验估计基础上运用能量方法分析了格式的稳定性及二阶收敛性．     

高精度隐式参差光滑格式的构造

参照Lax-Wendroff格式的构造方法,就双曲型方程、抛物型方程和双曲-抛物型方程,构造了一种新的IRS(implicit residual smoothing)格式。该IRS格式有二阶或三阶时间精度且大大地拓宽了解的稳定区域和CFL数。这种新的IRS格式有中心加权型和迎风偏向型两种,并用von-Neumann分析方法分析了格式的稳定范围。讨论了在透平机械中广泛应用的Dawes方法的局限性,发现该方法对稳态问题得出的解与时间步长的选取有关,对粘性问题求解时,时间步长受严格限制。最后,结合TVD(total variation diminishing)格式和四阶Runge-Kutta技术,用IRS格式和Dawes方法对二维反射激波场进行了数值模拟,数值结果支持本文的分析结论。     

二维抛物型方程的高精度多重网格解法

提出了数值求解二维抛物型方程的一种新的高精度加权平均紧隐格式，利用Fourier分析方法证明了该格式是无条件稳定的，为了克服传统迭代法在求解隐格式是收敛速度慢的缺陷，利用了多重网格加速技术，大大加快了迭代收敛速度，提高了求解效率，数值实验结果验证了方法的精确性和可靠性。     

守恒型双曲方程的中心差分TVD格式研究

研究了如何利用迎风格式的耗散性构造中心差分ＴＶＤ格式的方法，给 相应的定理，构造出新的耗散表达式。新格式既保留了二阶中心差分格式灵活方便的优点，又吸收了迎风格式耗散项比较精细的特点，同时具有ＴＶＤ性质，使得新格式具有较同的激波分辨率。     

一类时空二阶精度高分辨率MmB差分格式的构造及数值试验

1．引言考虑如下二维双曲型守恒律初值问题的数值解．H．M．Wu和S．L．Yang在文山中给出了MmB差分格式的定义如下：给定（．1）M差分格式定义．若则称格式（1．2）为MmB差分格式．这里BmB表示局部MaximumandminimumBounds．由定义可知，若差分格式（1．2）可写为形式且。＼P’三0，＞。：r’一1．则格式（1．4）为MmB差分格式．j＝l文山构造了二维双曲型守恒律的二类二阶精度的MmB差分格式，使构造二维高分辨格式有了新的突破，但他们是从标量线性双曲型守恒律出发，然后把结果推广到非线性情形．本文直接从二维非线性双曲型守恒律…     

Four-class skew-symmetric association schemes

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we investigate 4-class skew-symmetric association schemes. In recent work by the first author it was discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S.Y. Song in 1996. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that none of 2-class Johnson schemes admits a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.     

激波捕捉差分方法研究

在迎风型格式和矢通量分裂技术的基础之上,对捕捉激波方法进行一种新的尝试．该方法首先对原始格式在特征方向上进行投影,然后用限制器对这些特征分量的变化幅值进行限制以抑止非物理波动,最后再把它转换成守恒形式,得到了基本上无振荡的激波捕捉格式．用该方法对两种迎风显示格式(二阶和三阶)和3种迎风紧致格式(三阶、五阶和七阶)进行处理,并在一维和二维的情况下进行了应用测试．通过与高阶WENO、MP、Compact-WENO等格式的比较,表明该方法在光滑捕捉激波的前提下仍有较高精度和分辨率．     

线性传输方程的几种数值格式的比较

本文研究了线性传输方程的数值计算问题.利用Godunov格式、Entropy格式、Ultra-bee格式和Entropy-Ultra-bee格式对线性传输方程进行了数值计算,获得了相应的数值结果.数值实验结果表明Entropy-Ultra-bee格式结合了Entropy格式和Ultra-bee格式的优点,在整个计算区域都有比较高的分辨率,而且没有出现非物理振荡.     

一类TVD型的迎风紧致差分格式

给出一种迎风型TVD(total variation diminishing)格式的构造方法,该方法通过限制器来抑制线性紧致格式在模拟间断流场时的非物理波动,可构造出非线性TVD型紧致格式(CTVD)．然后采用该法构造出了3阶和5阶的TVD型紧致格式,并通过模拟一维组合波和Riemann问题,二维激波-涡相互干扰和激波-边界层相互作用等来考察它们的性能．数值实验表明了该类格式的高阶精度和分辨率,且过间断基本无振荡．     

Nilpotent schemes and group-like schemes

We give a definition of nilpotent association schemes as a generalization of nilpotent groups and investigate their basic properties. Moreover, for a group-like scheme, we characterize the nilpotency by its character products.     

The distribution of hermitian indefinite quadratic forms

Hermitian quadratic forms in complex Gaussian random variables is considered in this article. Various representations of the exact densities for the central, non central, singular and nonsingular cases are given. Particular cases are shown to agree with the results obtained by others in connection with various applications in communication theory, multichannel antennas, vehicular transportations and other related problems     

A category of association schemes

We define a category of association schemes and investigate its basic properties. We characterize monomorphisms and epimorphisms in our category. The category is not balanced. The category has kernels, cokernels, and epimorphic images. The category is not an exact category, but we consider exact sequences. Finally, we consider a full subcategory of our category and show that it is equivalent to the category of finite groups.     

Quot Functors for Deligne-Mumford Stacks

Abstract

Given a separated and locally finitely-presented Deligne-Mumford stack 𝒳 over an algebraic space S, and a locally finitely-presented 𝒪_𝒳-module ?, we prove that the Quot functor Quot(?/𝒳/S) is represented by a separated and locally finitely-presented algebraic space over S. Under additional hypotheses, we prove that the connected components of Quot(?/𝒳/S) are quasi-projective over S.     

An efficient and conservative hybrid method for solving multidimensional conservation laws

We have developed an efficient hybrid technique for solving nonlinear conservation laws. Previous hybrid techniques have been accurate but lacked the property of conservation, whereas our technique is both accurate and conservative. To achieve this, we superimposed all possible stencils for ENO polynomials and weighted the value from each cells in a way that depends on the numerical solution. Computational efficiency relies on switching from central data where the exact solution is smooth to noncentral data near discontinuities. We prove the theoretical consistency of the technique and discuss the connection with ENO and WENO methods. We introduce time dependency by combining our method with Runge‐Kutta schemes that are TVD preserving. We have verified our technique experimentally by solving a suite of test problems with convex and non‐convex flux functions taken from the literature. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009