三维多面体网格上扩散方程的保正格式

 针对三维任意(星形)多面体网格, 本文构造了扩散方程的一种单元中心型非线性有限体积格式, 证明了该格式具有保正性. 在该格式设计中, 除引入网格中心量外, 还引入网格节点量和网格面中心量作为中间未知量, 它们将用网格中心未知量线性组合表示, 使得格式仅有网格中心未知量作为基本未知量. 在节点量计算中, 利用网格面上的调和平均点, 设计了一种适用于三维多面体网格的局部显式加权方法. 该格式适用于求解非平面的网格表面和间断扩散系数的问题. 数值例子验证了它对光滑解具有二阶精度和保正性.     

大变形网格上时间二阶精度扩散方程的保正格式

本文基于已有的连续扩散通量的两点非线性离散格式,构造了2D非稳态扩散方程大变形网格上的两层非线性有限体积格式.该格式利用Crank-Nicolson (C-N)方法的思想在时间方向获得了二阶精度.由于所得代数方程组的系数矩阵的转置是M矩阵,从而能够保持解的正性,并利用Brouwer不动点定理证明了格式解的存在性.数值实验结果表明,在较大时间步长下,该格式具有二阶计算精度.     

不可压缩理想磁流体力学方程组的奇异极限

文中利用不可压缩理想磁流体力学（ＩＭＨＤ）方程组的特殊结构，引入变量代换，证明场论中的有关恒等式，克服失去双曲性所带来的困难，得到了一致能量估计，构造迭代序列，证明了当Ａｌｆｖｅｎ数０时，解的奇异极限．     

求解二维Fisher-KPP方程的一类保正保界差分格式及其Richardson外推法

本文研究求解二维Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP)方程的一类保正保界差分格式.运用能量分析法证明了当网格比满足$R_{x}+R_{y}+[b\tau (p-1)]/2\leq\frac{1}{2}$时差分解具有一系列数学性质,包括保正性、保界性和单调性,且在无穷范数意义下有$O (\tau+h_{x}^{2}+h_{y}^{2})$的收敛阶.然后通过发展Richardson外推法得到收敛阶为$O (\tau^{2}+h_{x}^{4}+h_{y}^{4})$的外推解.最后数值实验表明数值结果与理论结果相吻合.值得提及的是在运用本文构造的Richardson外推法时对时空网格比没有增加更严格的条件.     

L_2逼近高精度格式的基本原理及其应用

In the present paper, a new numerical method: L_2 approximation high accurate scheme is developed. The solution obtained by using this method satisfies not only at the discrete points, but also approximates to the exact solution in the total region. The basic principle is introduced and this method is used to solve some problems. The results show its high accuracy, high resolution and other advantages.     

一类满足熵增条件的流体力学方程守恒型格式

Lax,Wandtoff曾经证明:对于与守恒律方程组相容的守恒型差分格式,如果其差分解几乎处处有界收敛,那么极限函数是原方程组的一个弱解,并且提出了二阶精度的L-W格式.但是,一些数值计算表明,用二阶守恒型格式(如L-W格式及Mac Corma-ck格式),可能得到非物理解的计算结果.通常称满足熵条件的弱解为物理解.对     

流体力学方程组的总熵增量小的守恒型差分格式

1.引言 近年来,国外许多学者对求解双曲守恒律组的高分辨率、高精度差分格式进行了深入的研究。例如MUSCL方法、TVD格式、PPM方法、各种限流的方法以及ENO格式等等。将这些方法应用于流体力学方程组,其数值实践的结果表明,在消除波后振荡、提高激波间断分辨率、提高计算精度等方面有明显的效果。在设计这些     

扩散方程一种无条件稳定的保正并行有限差分方法

本文针对扩散方程提出了一种保正的并行差分格式,并且这个格式为无条件稳定的.我们在每个时间层将计算区域分成许多个子区域以便于实施并行计算.格式构造中首先我们使用前两个时间层的计算结果在分区界面处通过一种非线性的保正外插来预估子区域界面值.然后在每个子区域内部使用经典的全隐格式进行计算.最后在界面处使用全隐格式进行校正（本质上这一步计算是显式计算）.我们给出了一维与二维情形下的保正并行差分格式,并相应的给出了无条件稳定性证明.数值实验显示此并行格式具有二阶数值精度,而且无条件稳定性与保正性也均在数值实验中得到验证.     

抛物型方程的一个高精度隐格式

利用待定参数法,对一维抛物型方程构造出了一个截断误差为Ｏ（△ｘ＾４＋△ｘ＾４）的隐式差分格式,格式的稳定性条件为ｒ＝ａ△ｔ／△ｘ＾２≤１／√２,可用追赶法求解。     

SPECTRAL-DIFFERENCE METHOD FOR TWO-DIMENSIONAL COMPRESSIBLE FLUID FLOW

1.IntroductionWeconsiderthefollowingcompressibleflowequations:whereuisthevelocity,u~(ul,uZ)*,Tistheabsolutetemperature,u(T,p)isthviscouscoefficient,K(T,p)~Y'(T,p)--!.(T,p)withU'(T,p)beingthesecondviscoufOAScoefficient.P(T,p)isthecoefficientofheatconductio…     

ERROR ESTIMATION OF SPECTRAL-DIFFERENCE METHOD FOR COMPRESSIBLE FLUID FLOW IN THREE-DIMENSIONAL SPACE

This paper is devoted to a combined Fourier spectral-finite difference method for solving 3-dimensional, semi-periodic compressible fluid flow problem. The error estimation, as well as the convergence rate, is presented.     

A GHOST FLUID BASED FRONT TRACKING METHOD FOR MULTIMEDIUM COMPRESSIBLE FLOWS

Recent years the modify ghost fluid method （MGFM） and the real ghost fluid method （RGFM） based on Riemann problem have been developed for multimedium compressible flows. According to authors, these methods have only been used with the level set technique to track the interface. In this paper, we combine the MCFM and the RGFM respectively with front tracking method, for which the fluid interfaces are explicitly tracked by connected points. The method is tested with some one-dimensional problems, and its applicability is also studied. Furthermore, in order to capture the interface more accurately, especially for strong shock impacting on interface, a shock monitor is proposed to determine the initial states of the Riemann problem. The present method is applied to various one- dimensional problems involving strong shock-interface interaction. An extension of the present method to two dimension is also introduced and preliminary results are given.     

求解三维高次拉格朗日有限元方程的代数多重网格法

本文针对带有间断系数的三维椭圆问题,讨论任意四面体剖分下的二次拉格朗日有限元方程的代数多重网格法．通过分析线性和高次有限元空间之间的关系,我们给出了一种新的网格粗化算法和构造提升算子的代数途径．进一步,我们还对新的代数多重网格法给出了收敛性分析．数值实验表明这种代数多重网格法对求解二次拉格朗日有限元方程是健壮和有效的。     

HIGH－ORDER I－STABLE CENTERED DIFFERENCE SCHEMES FOR VISCOUS COMPRESSIBLE FLOWS

In this paper we present high-order I-stable centered difference schemes for the numerical simulation of viscous compressible flows.Here I-stability refers to time discretizations whose linear stability regions contain part of the imaginary axis,This class of schemes has a numerical stability independent of the cell-Reynolds number Rc,thus allows one to simulate high Reynolds number flows with relatively larger Rc,or coarser grids for a fixed Rc.On the other hand,Rc cannot be arbirarily large if one tries to obtain adequate numerical resolution of the viscous behavior,We investigate the behavior of hight-order I-stable schemes for Burgers‘ equation and the compressible Navier-Stokes equations.We demonstrate that,for the second order scheme,Rc≤3 is an appropriate constraint for numerical resolution of the viscous profile,while for the fourth-order schems the constraint can be relaxed to Rc≤6.Our study indicates that the fourth order order scheme is preferable:better accuracy,higher resolution,and larger cell-Reynolds numbers.     

高阶抛物型方程的一族高精度恒稳差分格式

A family of three-layer implicit difference Schemes of high accuracy with two parameters for solving high order parabolic equationδu/δt=(-1)^m 1δ^2mu/δx^2m(where m is positive integers) are constructed. In the special case α=1/2, β=0, We obtain a two-layer difference scheme. These schemes are proved to be absolutely stable for arbiratily chosen non-negative parameters, And the order of the truncation error is O((△t)^2 (△x)^6). They are shown by numerical examples to be effective, and practice consistant with theoretical analysis.     

四阶杆振动方程的两类高精度辛格式

In this paper, we present two classes of symplectic schemes with high order accuracy for solving four-order rod vibration equation utt uxxxx=0 via the third type generating function method. First, the equation of four order rod vibration is written into the canonical Hamilton system; second, overcoming successfully the essential difficult on the calculus of high order variations derivative, we get the semi-discretization with arbitrary order of accuracy in time direction for the PDEs by the third type generating function method. Furthermore the discretization of the related modified equation of original equation is obtained. Finally, arbitrary order accuracy symplectic schemes are obtained. Numerical results are also presented to show the effectiveness of the scheme, high order accuracy and properties of excellent long-time numerical behavior.     

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…     

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.     

一维高精度离散GDQ方法

GDQ method is a kind of high order accurate numerical methods developed several years ago, which have been successfully used to simulate the solution of smooth engineering problems such as structure mechanics and incompressible fluid dynamics. In this paper, extending the traditional GDQ method, we develop a new kind of discontinuous GDQ methods to solve compressible flow problems of which solutions may be discontinuous. In order to capture the local features of fluid flows, firstly, the computational domain is divided into many small pieces of subdomains. Then, in each small subdomain, the GDQ method is implementedand some kinds of numerical flux limitation conditions will be required to keep the correct flow direction. At the boundary interface between subdomains, we also use some kind of flux conditions according to the flow direction. The numerical method obtained by the above steps has the advantages of high order accuracy and easy to treat boundary conditions. It can simulate perfectly nonlinear waves such as shock, rarefaction wave and contact discontinuity. Finally, the numerical experiments on one dimensional Burgers equation and Euler equations are given.The numerical results verify the validation of the method.