异步并行非线性对称Gauss-Seidel迭代算法

1．引言考虑非线性方程组其中A＝（a。。）EL（＊”）为＊一矩阵，B＝（衬。）EL（＊”）为非负矩阵，呐X）一（p。（X。》，4（二）＝（吵k（kk》：＊一＊一为连续的对角映射，而6＝（6k）E＊一为已知向量．这里，什小：”一”均可微，但二者的导函数并不一定连续．这类方程组具有丰富的实际背景．例如，描述冰体溶解过程的著名的Stefan问题，就可归结为问题（1·1）的数值求解（见［l］）．为在多处理机系统上有效地求解问题（1．1），文山利用这类非线性方程组的特殊结构，建立了一类并行非线性Gauss—Seidel型迭代算法．为避免该算…     

非线性波动方程的交替显-隐差分方法

1．引言众所周知，非线性波动方程在自然科学领域有广泛的物理背景，诸如物理、化学反应方程，机械动力学方程，地球物理与大气海洋方程等．差分方法求解非线性波动方程已有研究，如［1］和IZ］就给出了非线性波动方程组的显式和隐式差分格式以及收敛性分析．虽然古典的显式差分格式易于并行计算，但是它的稳定性条件差（条件稳定）；古典的隐式差分格式稳定性条件好（绝对稳定）；但对非线性问题，一般需要线性化，然后求解一个线性代数方程组，并行计算能力差．本文正是在这样一种前题下，给出了一维问题的一种交替分段显一隐差分格式，…     

同位非结构和结构网格摄动有限体积法（PFV）求解二维Navier-Stokes方程组

根据NS方程组的一阶迎风和二阶中心有限体积（UFV和CFV）格式,导出NS方程组迎风和中心摄动有限体积(UPFV和CPFV)格式．该格式通过把控制体界面质量通量摄动展开成网格间距的幂级数,并由守恒方程本身求得幂级数系数而获得．迎风和中心摄动有限体积格式使用了与一阶迎风和二阶中心格式相同的基点数和相同的表达形式,宜于计算机编程．顶盖驱动方腔流和驻点流标量输运的数值实验证明,迎风PFV格式比一阶UFV、二阶CFV格式有更高的精度,更高的分辨率．尤其是良好的鲁棒特性．对顶盖驱动方腔流,在Re数从102到104范围内,亚松弛系数可在0.3～0.8任取,收敛性能良好．     

广义块Broyden方法与超定方程组求解

1．引言［1］提出用块Broyden方法求解成组的线性与非线性方程组，同时证明了：若有p组n阶线性方程组则块Brorden方法具有至多2n／p步的有限终止性．这种块形式算法，对于大型成组问题的计算，在计算量和存储量方面，都会有相当的改善，并且有利于并行计算．本文推广上述结果，建立一种广义块nroxaen方法，并将它应用于成组的超定方程组的求解．我们证明了对于给出的p组。x叫x三叫的线性超定方程组其中AeRm””，x；e＊”，kEBm，广义块Broxden方法同样具有至多z。／r步的有限终止性，这表明超定方程组的纽数越多（P5…，方法所需的选代…     

可显式求解二维扩散方程组的三层差分格式

1．问题的提出在区域D三队1）x阻1）x叮n考虑二线扩散方程组的初边值问题：的差分解．假定系数人心民小。,q二1,…,川满足条件KI～＊3：KI．A。P（x,yt）二AP。（x,y,t）,B。P（x,yt）＝BO。（x,y,t）;KZ．存在常数。1,。2＞0使得对于V（2,从t）ED八佰,…,（M）ERM,有如下不等式成立：＊3．人p和民p在D二队则x叮11x叮n上充分光滑,且关于土为h耶．连续,于是存在常数K＞0,使得对于一维清形,陈光南山采用了一种三层主对角隐格式,通过引入高阶人工粘性项使格式绝对稳定,沈隆钩等［2,3］在研究Shrsdinger型方程…     

解非定常不可压Navier-Stokes方程的压力修正投影法的并行算法研究

1．CNMTZ格式和压力Poisson、方程的快速解法为后面讨论的方便，本节中我们简要介绍一下［1，2］提出的求解上述方程的CNMTZ算法以及相应的压力Poisson方程的快速算法．考虑原始变量非定常不可压N—S方程（INSE）和满足方程（2的初始条件W（；队0），W在边界＊0上给定并满足这里，V＝（。，V厂是速度场，p是压力．方程（1）（2）进行空间离散后可以写成以下形式：其中力（W，t）包括动量方程中的对流项、扩散项和非齐次项的空间差分近似及相应的边界条件，G和D分别为梯度和散度算子的离散形式．它的CNMT格式的PC投影法如下：其…     

线性流形上矩阵方程AX=B的一类反问题及数值解法

1．引言本文用＊-"m表示全体nX。实矩阵的集合，人表示n阶单位矩阵，汉"m一《ME＊""叫rank（川一r），＊＊"""＝HE＊"""卜"A＝v，＊＊"""一仰E＊"""卜"一M｝，SR；""（SR7""）表示全体7。阶实对称半正定（正定）阵集合．N（A）表示矩阵A的零空间，即N（A）＝（xlAx＝0），ID叫D表示Frobenius范数，A"表示矩阵A的Moors－Penrose广义逆，［EI十表示在Frobenius范数意义下n阶方阵E在SR；""中唯一的最佳k逼近解，即口一［E］＋11-inf。。、。。x，IllE-All．（［E］十求法见文［7］）．还用A三0（A三0）表示A（的k阶顺序主子矩…     

波动方程的一类显式辛格式

1．引言和预备知识本文主要考虑如下波动方程初边值问题的数值方法．一般的哈密顿系统可写成那么，系统（1．幻称为可分的哈密顿系统．众所周知，方程（1．la）在引进新变量。t＝v后，它变成一类可分的线性哈密顿系统：它的哈密顿函数为H一三矿（V“＋。乏）咖，并且在离散＊。＝。。。后，（1．4）是一个方程组，u，V是向量．人们早就知道，初边值问题（1．1）在应用隐式中点公式（辛格式）进行数值解时，能保持真解许多重要性质，并且格式是无条件稳定的，但美中不足的是，在每积分步，它要求解一个线性方程组，当考虑大的离散系统时，…     

对流扩散方程一类改进的特征线修正有限元方法

１引言在地下水污染,地下渗流驱动,核污染,半导体等问题的数值模拟中,均涉及抛物型对流扩散方程（或方程组）的数值求解问题．这些对流扩散型偏微分方程（或方程组）具有共同的特点：对流的影响远大于扩散的影响,即对流占优性,对流占优性给问题的数值求解带来许多困难,因此对流占优问题的有效数值解法一直是计算数学中重要的研究内容．用通常的差分法或有限元法进行数值求解将出现数值振荡．为了克服数值振荡,提出各种迎风方法和修正的特征方法并在这些问题上得到成功的实际应用、８０年代,Ｄｏｕｇｌａｓ和Ｒｕｓｓｅｌｌ［２］等…     

关于离散代数Riccati方程的扰动分析

1．引言若不特别说明,以下记号都是常规的,可参见山．在最优控制中占有核心地位的代数Xiccati方程（ARE）有两种基本形式：连续型的ARE（CARE）：离散型的＊＊E（＊＊RE）：其中只兄见NE贮””,GIE皿””m,GZE＊m”m,in＜2,＊T二K＞风＊T二N＞几＊2二＊2＞凡从应用角度看,主要关心＊＊E的对称半正定解．对于＊＊RE,已有大量的研究工作．特别,陈春晖门、徐树方问、Ghwimi－Laub同等研究了扰动理论．对于DARE,它与＊＊＊E的一个明显区别是：＊＊M是二次矩阵方程而＊＊＊E具有高度非线性．这种区别使得DARE远为复杂…     

KINETIC FLUX VECTOR SPLITTING FOR THE EULER EQUATIONS WITH GENERAL PRESSURE LAWS

This paper attempts to develop kinetic flux vector splitting(KFVS)for the Euler equa-tions with general pressure laws.It is well known that the gas distribution function forthe local equilibrium state plays an important role in the construction of the gas-kineticschemes.To recover the Euler equations with a general equation of state(EOS),a newlocal equilibrium distribution is introduced with two parameters of temperature approx-imation decided uniquely by macroscopic variables.Utilizing the well-known connectionthat the Euler equations of motion are the moments of the Boltzmann equation wheneverthe velocity distribution function is a local equilibrium state,a class of high resolutionMUSCL-type KFVS schemes are presented to approximate the Euler equations of gas dy-namics with a general EOS.The schemes are finally applied to several test problems for ageneral EOS.In comparison with the exact solutions,our schemes give correct location andmore accurate resolution of discontinuities.The extension of our idea to multidimensionalcase is natural.     

Jacobian系数矩阵分裂法与气动方程的数值计算

前言 七十年代中期,人们多采用显式方法数值求解可压缩的Navier-Stokes方程.这种方法简单易解,但由于稳定性对时间步长的限制,使得求解所需机时颇多.在求解定常问题时,数值求解过程可以与真实的物理发展过程不对应,人们可以根据需要而改变求解过程,以达到加速收敛的目的.Allen和Cheng就是根据这种思想计算了近底部分离流动.为了达到加速得到定常解的目的,很多人采用在不同空间点上取变时间步长的方法.在[5]中,当调节因子取标量形式时,相当于取变时间步长的方法.如果调节因子或算子放大修正系数取矩阵形式,则可得到更快的收敛速度.Beam和Warming在[7]中提出了一个非迭代的隐式方法,并在空间坐标方向上利用近似因式分解,大大提高了隐式格式的使用效率.Steger和Warming在[8]中详细介绍了流通量分裂法.1985年,MacCormack在[9]中改进了自己在[10]中提出的二步隐式方法.作者在[11]中也用流     

The unconditional stability and mass‐preserving S‐DDM scheme for solving parabolic equations in three dimensions

In this paper, the unconditional stability and mass‐preserving splitting domain decomposition method (S‐DDM) for solving three‐dimensional parabolic equations is analyzed. At each time step level, three steps (x‐direction, y‐direction, and z‐direction) are proposed to compute the solutions on each sub‐domains. The interface fluxes are first predicted by the semi‐implicit flux schemes. Second, the interior solutions and fluxes are computed by the splitting implicit solution and flux coupled schemes. Last, we recompute the interface fluxes by the explicit schemes. Due to the introduced z‐directional splitting and domain decomposition, the analysis of stability and convergence is scarcely evident and quite difficult. By some mathematical technique and auxiliary lemmas, we prove strictly our scheme meet unconditional stability and give the error estimates in L²‐norm. Numerical experiments are presented to illustrate the theoretical analysis.     

Steger–Warming flux vector splitting method for special relativistic hydrodynamics

This paper discusses the properties of the rotational invariance and hyperbolicity in time of the governing equations of the ideal special relativistic hydrodynamics and proves for the first time that the ideal relativistic hydrodynamical equations satisfy the homogeneity property, which is the footstone of the Steger–Warming flux vector splitting method [J. L. Steger and R. F. Warming, J. Comput. Phys., 40(1981), 263–293]. On the basis of this remarkable property, the Steger–Warming flux vector splitting (SW‐FVS) is given. Two high‐resolution SW‐FVS schemes are also given on the basis of the initial reconstructions of the solutions and the fluxes, respectively. Several numerical experiments are conducted to validate the performance of the SW‐FVS method. Copyright © 2013 John Wiley & Sons, Ltd.     

Flux-splitting schemes for parabolic equations with mixed derivatives

Difference schemes of required quality are often difficult to construct as applied to boundary value problems for parabolic equations with mixed derivatives. Specifically, difficulties arise in the design of monotone difference schemes and unconditionally stable locally one-dimensional splitting schemes. In parabolic problems, certain opportunities are offered by restating the problem in question so that the quantities to be determined are fluxes (directional derivatives). The original problem is then rewritten as a boundary value one for a system of equations in flux variables. Weighted schemes for parabolic equations in flux coordinates are examined. Unconditionally stable locally one-dimensional flux schemes that are first- and second-order accurate in time are constructed for a parabolic equation without mixed derivatives. A feature of systems in flux variables for equations with mixed derivatives is that the terms with time derivatives are coupled with each other.     

High-order accurate monotone compact running scheme for multidimensional hyperbolic equations

Monotone absolutely stable conservative difference schemes intended for solving quasilinear multidimensional hyperbolic equations are described. For sufficiently smooth solutions, the schemes are fourth-order accurate in each spatial direction and can be used in a wide range of local Courant numbers. The order of accuracy in time varies from the third for the smooth parts of the solution to the first near discontinuities. This is achieved by choosing special weighting coefficients that depend locally on the solution. The presented schemes are numerically efficient thanks to the simple two-diagonal (or block two-diagonal) structure of the matrix to be inverted. First the schemes are applied to system of nonlinear multidimensional conservation laws. The choice of optimal weighting coefficients for the schemes of variable order of accuracy in time and flux splitting is discussed in detail. The capabilities of the schemes are demonstrated by computing well-known two-dimensional Riemann problems for gasdynamic equations with a complex shock wave structure.     

Three-level schemes of the alternating triangular method

In this paper, the schemes of the alternating triangular method are set out in the class of splitting methods used for the approximate solution of Cauchy problems for evolutionary problems. These schemes are based on splitting the problem operator into two operators that are conjugate transposes of each other. Economical schemes for the numerical solution of boundary value problems for parabolic equations are designed on the basis of an explicit-implicit splitting of the problem operator. The alternating triangular method is also of interest for the construction of numerical algorithms that solve boundary value problems for systems of partial differential equations and vector systems. The conventional schemes of the alternating triangular method used for first-order evolutionary equations are two-level ones. The approximation properties of such splitting methods can be improved by transiting to three-level schemes. Their construction is based on a general principle for improving the properties of difference schemes, namely, on the regularization principle of A.A. Samarskii. The analysis conducted in this paper is based on the general stability (or correctness) theory of operator-difference schemes.     

The Runge‐Kutta DG finite element method and the KFVS scheme for compressible flow simulations

This article presents a new type of second‐order scheme for solving the system of Euler equations, which combines the Runge‐Kutta discontinuous Galerkin (DG) finite element method and the kinetic flux vector splitting (KFVS) scheme. We first discretize the Euler equations in space with the DG method and then the resulting system from the method‐of‐lines approach will be discretized using a Runge‐Kutta method. Finally, a second‐order KFVS method is used to construct the numerical flux. The proposed scheme preserves the main advantages of the DG finite element method including its flexibility in handling irregular solution domains and in parallelization. The efficiency and effectiveness of the proposed method are illustrated by several numerical examples in one‐ and two‐dimensional spaces. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

一维气液两相漂移模型全隐式AUSMV算法研究

气液两相漂移模型显式AUSMV（advection upstream splitting method combined with flux vector splitting method）算法的时间步长受限于CFL（Courant-Friedrichs-Lewy）条件，为了提高计算效率，建立了一种全隐式AUSMV算法求解气液两相漂移模型.采用AUSM格式结合FVS（flux vector splitting）格式构造连续方程和运动方程的对流项数值通量，AUSM格式构造压力项数值通量.离散控制方程是非线性方程组，采用六阶Newton(牛顿)法结合数值Jacobi矩阵求解.计算经典算例Zuber-Findlay激波管问题和复杂漂移关系变质量流动问题，结果分析表明:全隐式AUSMV算法，色散效应小，无数值震荡，计算精度高.在压力波波速高的条件下，可以显著提高计算效率，耗散效应小.     

Flux-splitting schemes for parabolic problems

Splitting with respect to space variables can be used in solving boundary value problems for second-order parabolic equations. Classical alternating direction methods and locally one-dimensional schemes could be examples of this approach. For problems with rapidly varying coefficients, a convenient tool is the use of fluxes (directional derivatives) as independent variables. The original equation is written as a system in which not only the desired solution but also directional derivatives (fluxes) are unknowns. In this paper, locally one-dimensional additional schemes (splitting schemes) for second-order parabolic equations are examined. By writing the original equation in flux variables, certain two-level locally one-dimensional schemes are derived. The unconditional stability of locally one-dimensional flux schemes of the first and second approximation order with respect to time is proved.