双曲型守恒律方程组的Godunov格式中离散激波的渐近稳定性

１引言我们研究求解守恒律初值问题的Ｇｏｄｕｎｏｖ格式,其中ｆ：ＲＮ→ＲＮ,并且方程（１）是严格双曲型的．我们还假定　的特征值（）,…,Ｎ（）均为非零,并且每一个特征在Ｌａｘ意义下或者是真正非线性的,或者是线性退化的．利用Ｍａｊｄａ和Ｒａｌｓｔｏｎ的一个结果［６］,我们将证明离散行波激波的存在性,然后证明激波的渐近稳定性,也即,如果初值是激波的一个小的摄动,则当ｔ→由Ｇｏｄｕｎｏｖ格式得到的解趋于一个平移了的行波激波．在［１０］中我们证明了对于Ｌａｘ－Ｆｒｉｅｄｒｉｃｈｓ格式的渐近稳定性,所用的加…     

一个求解多维守恒律方程组的二阶显式有限元格式

１．引言 近年来,在非结构网格上求解双曲型守恒律的数值方法引起了较为广泛的关注,出现了有限体积方法［１］,间断 Ｇａｌｅｒｋｉｎ方法 ［２］,流线扩散方法［３］,以及 ＮＮＤ格式 ［４］等．我们在［６,７］中提出了一种求解双曲型守恒律方程式的有限元方法,它是在一个求解对流扩散问题的有限元方法 ［５］的基础上发展起来的．它是一个显式有限元方法,因此计算量很小．在这个方法中,我们将任意维的问题归结为在单元棱边上的一维计算,引入了积分因子,因此在单元内部可以容纳边界层．这样,它特别适合于对流占优问题以及双曲…     

一维双曲守恒方程组的Taylor-Galerkin有限元方法

１．引言 在文章［８］中,利用双曲守恒律的Ｈａｍｉｌｔｏｎ－Ｊａｃｏｂｉ方程形式,应用 Ｇａｌｅｒｋｉｎ有限元给出了求解一维双曲守恒律的计算方法．不同于间断有限元方法［２］、［３］和 Ｔａｙｌｏｒ－Ｇａｌｅｒｋｉｎ有限元方法［１］求解双曲守恒律,文章［８］采用连续 Ｇａｌｅｒｋｉｎ有限元求解双曲守恒律． 在文章［８］中,对差分方法和有限元方法求解双曲守恒律作了较为详细的讨论．同时在文章［８］中,采用积分变换,将双曲守恒律方程变成 Ｈａｍｉｌｔｏｎ－Ｊａｃｏｂｉ方程形式．由于 Ｈａｍｉｌｔｏｎ－Ｊａｃｏ…     

基于近似惯性流形的后验Galerkin的方法

１．引言 为提高用数值方法解非线性发展方程及非线性椭圆边值问题的逼近阶,许多学者例如Ｊ．Ｎｏｖｏ和 Ｅ．Ｔｉｔｉ［４］, Ｍａｒｉｏｎ和 Ｔｅｍａｎ［６］,Ｊ．Ｘｕ［７］以及 Ｗ．Ｌａｙｔｏｎ［９］等人,提出了后验Ｇａｌｅｒｋｉｎ方法、近似惯性流形方法、非线性Ｇａｌｅｒｋｉｎ方法、各种区域分裂法、多重网格法等等．本文根据［１］提出了一种新的高精度的后验 Ｇａｌｅｒｋｉｎ方法．它的逼近阶是经典 Ｇａｌｅｒｋｉｎ方法逼近阶的两倍． 考虑非线性椭圆边值问题这里ｎ是按ｄ＝２,３）上具有分段光滑边界ｒ的有界区域,…     

形状记忆合金问题的有限元逼近

１．引言本文讨论非线性微分方程其中　系数　是给定常数,ｆ,ｆ为已知函数．这是形状记忆合金问题的数学模型,未知量ｕ,θ代表位移及Ｋｅｌｖｉｎ温度,其物理背景及数学模型的建立,参见文献［３,４］．最近,文［１,２］讨论了方程组（１．１）－（１．５）的数值求解,提出全离散格式．文［２］用Ｇａｌｅｒｋｉｎ方法,位移ｕ用四阶微分方程的有限个特征向量张成的空间,温度θ用分段线性多项式（折线）空间来近似,给出一个全离散格式,证明了离散近似解的存在唯一性,定性说明收敛于原问题的精确解．文［１］采用［２］中的离散…     

一类带波动算子的非线性Schrodinger方程的一个守恒差分格式

该文对一类带波动算子的非线性Ｓｃｈｒ¨ｏｄｉｎｇｅｒ（ＮＬＳ）方程提出了一个守恒的差分格式，证明了该格式的收敛性和稳定性．数值计算结果表明，该格式对网比不敏感，具有很好的守恒性，并且比文［１］中的不守恒格式提高了计算效率．     

离散速度动力学方程组的数值方法研究 Ⅰ.半隐式差分格式

１．引言文中考虑 Ｂｏｌｔｚｍａｎｎ方程的离散速度模型一两速度模型的数值方法的构造和分析．其中,ｃ为分子速度,Ｊ（ｕ,ｖ）为碰撞算子,具有如下一般形式式中ｋ（ｕ,ｖ）,和为非负实常数．模型（１．１）－（１．２０包含了一些著名的动力学模型,例如 Ｇｏｌｄｓｔｅｉｎ－Ｔａｙｌｏｒ模型［５,１４］, Ｒｕｉｊｇｒｏｏｋ－Ｗｕ模型［１２］,多孔介质方程［９,１１］等． 如果引进宏观变量产＝ｕ＋ｖ,ｊ＝（ｕ－ｖ）,则可得到宏观方程其中此外,方程（１．３）有两个自然渐近区．第一个是此时Ｊ（ｊ）＝０,如果该方程有…     

一般形式的一阶椭圆型偏微分方程组拟线性Riemann-Hilbert问题

在文［ｌ，２，３］中，Ｅ．Ｗｅｇｅｒｔ和Ｌ．Ｖ．Ｗｏｌｆｅｒｓｄｏｒｆ等人讨论了一类全纯函数的拟线性Ｒｉｅｍａｎｎ－Ｈｉｌｂｅｒｔ 问题在 Ｈａｒｄｙ空间中的可解性，在文［４］中，讨论了广义解析函数的拟线性 Ｒｉｅｍａｎｎ－Ｈｉｌｂｅｒｔ问题，同样得到该边值问题在Ｈ２类解空间中的可解性、本文在前面研究工作的基础上，对一般形式的一阶椭圆型偏微分方程组拟线性Ｒｉｅｍａｎｎ－Ｈｉｌｂｅｒｔ问题作了更深入的讨论，在适当的假设条件下，应用积分算子理论，函数论方法及不动点原理，证明了该边值问题在相应的泛函空间中同样是可解的．     

对流扩散方程的经济差分格式

１．引言 对流扩散方程是一类基本的运动方程,它可描述质量、热量的输运过程以及反应扩散过程等众多物理现象．寻找稳定、快速实用的数值方法,有着重要的理论和实际意义．标准的差分方法或有限元方法对它常常失效,根本原因在于“对流项”的存在．［１］提出了解对流扩散方程的特征线修正技术,这一方法考虑沿着特征线（流动方向）的离散,利用了对流扩散问题的物理力学性质,可以有效地克服数值振荡,保证数值解的稳定,尤其对“对流占优”的问题,这一方法有突出的优越性．这方面已有大量的理论和应用研究成果［２,３,７］．对大规模…     

ARMA序列MA参数G—M估计的强相合性

文献［１］给出了ＡＲＭＡ序列ＭＡ参数的Ｇ－Ｍ估计,并证明了估计的渐近正态性．本文证明了这种估计的强相合性．     

一类通矢量分裂方法的保正性研究Ⅰ.显式格式

This paper is about the positivity analysis of a class of flux-vector splitting (FVS) methods for the compressible Euler equations, which include gas-kinetic Beam scheme[8], Steger-Warming FVS method[9], and Lax-Friedrichs scheme. It shows that the density and the internal energy could keep non-negative values under the CFL condition for all above three schemes once the initial gas stays in a physically realizable state. The proof of positivity is closely related to the pseudo-particle representation of FVS schemes.     

A construction of non-isomorphic amorphous association schemes from pseudo-cyclic association schemes

As a continuation of my paper [T. Fujisaki, A construction of amorphous association scheme from a pseudo-cyclic association scheme, Discrete Math. 285(1–3) (2004) 307–311], we show a construction of amorphous association scheme which is a fusion scheme of a direct product of two pseudo-cyclic association schemes with same first eigenmatrix. By using this construction, we can get at most three amorphous association scheme. We prove that if two pseudo-cyclic association scheme are non-isomorphic, then these three amorphous association schemes are mutually non-isomorphic.     

扩散方程保正的有限体积格式

在大变形网格上数值求解多介质扩散方程时, 如何构造具有保正性的扩散格式一直是人们关注的难题. 本文将简要综述与保正性相关的扩散格式的研究历史, 并为解决这一难题提出新的设计途径,构造出新的具有较高精度的单元中心型守恒保正格式, 它们可兼顾网格几何变形和物理量变化. 本文将给出数值实验结果, 验证新格式在变形的网格上保持非负性.     

Modified form of binary and ternary 3-point subdivision schemes

Binary 3-point scheme, developed by Hormann and Sabin [Hormann, K. and Sabin, Malcolm A., 2008, A family of subdivision schemes with cubic precision, Computer Aided Geometric Design, 25, 41-52], has been modified by introducing a tension parameter which generates a family of C¹ limiting curves for certain range of tension parameter. Ternary 3-point scheme, introduced by Siddiqi and Rehan [Siddiqi, Shahid S. and Rehan, K., 2009, A ternary three point scheme for curve designing, International Journal of Computer Mathematics, In Press, DOI: 10.1080/00207160802428220], has also been modified by introducing a tension parameter which generates family of C¹ and C² limiting curves for certain range of tension parameter. Laurent polynomial method is used to investigate the continuity of the subdivision schemes. The performance of modified schemes has been demonstrated by considering different examples along with its comparison with the established subdivision schemes.     

Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes

In this paper, numerical solution of the Burgers–Huxley (BH) equation is presented based on the nonstandard finite difference (NSFD) scheme. At first, two exact finite difference schemes for BH equation obtained. Moreover an NSFD scheme is presented for this equation. The positivity, boundedness and local truncation error of the scheme are discussed. Finally, the numerical results of the proposed method with those of some available methods compared.     

Euler-type schemes for weakly coupled forward-backward stochastic differential equations and optimal convergence analysis

We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations (FBSDEs). Although the schemes share some common features with the ones proposed by C. Bender and J. Zhang [Ann. Appl. Probab., 2008, 18: 143–177], less computational work is needed for our method. For both our schemes and the ones proposed by Bender and Zhang, we rigorously obtain first-order error estimates, which improve the half-order error estimates of Bender and Zhang. Moreover, numerical tests are given to demonstrate the first-order accuracy of the schemes.     

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.     

A vertex‐centered and positivity‐preserving scheme for anisotropic diffusion equations on general polyhedral meshes

We propose a new nonlinear positivity‐preserving finite volume scheme for anisotropic diffusion problems on general polyhedral meshes with possibly nonplanar faces. The scheme is a vertex‐centered one where the edge‐centered, face‐centered, and cell‐centered unknowns are treated as auxiliary ones that can be computed by simple second‐order and positivity‐preserving interpolation algorithms. Different from most existing positivity‐preserving schemes, the presented scheme is based on a special nonlinear two‐point flux approximation that has a fixed stencil and does not require the convex decomposition of the co‐normal. More interesting is that the flux discretization is actually performed on a fixed tetrahedral subcell of the primary cell, which makes the scheme very easy to be implemented on polyhedral meshes with star‐shaped cells. Moreover, it is suitable for polyhedral meshes with nonplanar faces, and it does not suffer the so‐called numerical heat‐barrier issue. The truncation error is analyzed rigorously, while the Picard method and its Anderson acceleration are used for the solution of the resulting nonlinear system. Numerical experiments are also provided to demonstrate the second‐order accuracy and well positivity of the numerical solution for heterogeneous and anisotropic diffusion problems on severely distorted grids.     

A three-time-level explicit difference scheme of the second order of accuracy for parabolic equations

The difference schemes of Richardson [1] and of Crank-Nicolson [2] are schemes providing second-order approximation. Richardson's three-time-level difference scheme is explicit but unstable and the Crank-Nicolson two-time-level difference scheme is stable but implicit. Explicit numerical methods are preferable for parallel computations. In this paper, an explicit three-time-level difference scheme of the second order of accuracy is constructed for parabolic equations by combining Richardson's scheme with that of Crank-Nicolson. Restrictions on the time step required for the stability of the proposed difference scheme are similar to those that are necessary for the stability of the two-time-level explicit difference scheme, but the former are slightly less onerous.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 751–759, November, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00489 and by the International Science Foundation under grants No. N8Q300 and No. JBR100.     

ON THE EXPLICIT COMPACT SCHEMES II: EXTENSION OF THE STCE/CE METHOD ON NONSTAGGERED GRIDS ON THE EXPLICIT COMPACT SCHEMES II: EXTENSION OF THE STCE/CE METHOD ON NONSTAGGERED GRIDS

1. IntroductionTills paper is interested in the genuinely nonlinear conserVation lawswith initial data u(0, x) = "o(x), x = (x', ...l x').It is well known that the above problem may not always have a smooth global solutinn even if the initial data no is adequately smooth[6]. Thus, we consider its weaksolutinn so that the Problem (1.1) Ililght have a global solution allowing discontinuities(e.g. shock wave.etc.). Moreover, the elltrOPy conditinn should be deposed inorder to single out a phyS…