CONSERVATION OF THREE—POINT COMPACT SCHEMES ON SINGLE AND MULTIBLOCK PATCHED GRIDS FOR HYPERBOLIC PROBLEMS

For nonlinear hyperbloic problems,Conservation of the numerical scheme is important for convergence to the correct weak solutions.In this paper the the conservation of the well-known compact scheme up to fourth order of accuracy on a single and uniform grid is studied,and a conservative interface treatment is derived for compact schemes on patched grids .For a pure initial value problem,the compact scheme is shown to be equivalent to a scheme in the usual conservative form .For the case of a mixed initial boundary value problem,the compact scheme is conservative only if the rounding errors are small enough.For a pactched grid interface,a conservative interface condition useful for mesh fefiement and for parallel computation is derived and its order of local accuracy is analyzed.     

一类通矢量分裂方法的保正性研究Ⅰ.显式格式

１．引 言 近几年,流体力学数值方法的保正性一直受到人们的关注,特别是在高速流或低密度流体的数值模拟中,具有保正性的数值方法就显得更重要,因为用它们计算的密度或内能不出现负值．Ｅｉｎｆｅｌｄｔ等人［１］首先研究了Ｇｏｄｕｎｏｖ－型方法在密度较低时的特性．他们证明了Ｇｏｄｕｎｏｖ格式［２］是正守恒的,而 Ｒｏｅ的近似 Ｒｉｅｍａｎｎ方法［７］却不是．他们也证明了Ｈａｒｔｅｎ等人的近似Ｒｉｅｍａｎｎ方法［４］也是正守恒的,如果最大和最小波速的绝对值满足一定的稳定性界．最近,Ｔａｎｇ和 Ｘｕ［１１］分析了…     

OPTIMAL POINT-WISE ERROR ESTIMATE OF A COMPACT FINITE DIFFERENCE SCHEME FOR THE COUPLED NONLINEAR SCHRODINGER EQUATIONS

In this paper, we analyze a compact finite difference scheme for computing a coupled nonlinear SchrSdinger equation. The proposed scheme not only conserves the totM mass and energy in the discrete level but also is decoupled and linearized in practical computa- tion. Due to the difficulty caused by compact difference on the nonlinear term, it is very hard to obtain the optimal error estimate without any restriction on the grid ratio. In order to overcome the difficulty, we transform the compact difference scheme into a special and equivalent vector form, then use the energy method and some important lemmas to obtain the optimal convergent rate, without any restriction on the grid ratio, at the order of O（h4 ＋r2） in the discrete L∞ -norm with time step - and mesh size h. Finally, numerical results are reported to test our theoretical results of the proposed scheme.     

广义sine-Gordon方程高精度隐式紧致差分方法

本文研究一类二维非线性的广义sine-Gordon(简称SG)方程的有限差分格式.首先构造三层时间的紧致交替方向隐式差分格式,并用能量分析法证明格式具有二阶时间精度和四阶空间精度.然后应用改进的Richardson外推算法将时间精度提高到四阶.最后,数值算例证实改进后的算法在空间和时间上均达到四阶精度.     

ON THE CONVERGENCE OF IMPLICIT DIFFERENCE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS

1. IntroductionWe are are interested in the fOllowing Cauchy problem for scalar conservation lawswhere the initial data uo E BV(R) and the flux function f 6 C'(n).It is well known that this problem may not always have a smooth global solution even ifthe i…     

ON THE CELL ENTROPY INEQUALITY FOR THE FULLY DISCRETE RELAXING SCHEMES

1. IntroductionThis paPer is interested in studies of the cell eatroPy ineqallty for two classes of the fullydiscrete relaxin schemes aPprchating the fOlfowing scalar consendion lawwith dritial data u(0, x) = bo(x), x = (x1, ...) xd).lt is wen knOWn that the above cauchy problem (1.1) may not ahas have a smooth globalsollltion even thOugh the initial data uo is smoOth [8, 9]. Thus, we consider its weak solutiOn sothat the prObem (1.1) ndght hav a global solution allOwing disconinulies (e.g…     

ON ENTROPY CONDITIONS OF HIGH RESOLUTION SCHEMES FOR SCALAR CONSERVATION LAWS

1. IntroductionLet us consider the Cauchy problems for nonlinear hroerbolic scalar cO~ion laws:where the function f(u) E C'(R) and the hatal data ho E BV(R). As is well lmown, thisProblem in general does not ~ smooth solution, so that weak solution in the sense ofdistributions must be considered. Moreover, an elitropy condition must be added in order toensure the ~s of the weak solutions.The research of numerical methods for solving the equation (1.1) has been developed rapidlyin this deca…     

HIGH RESOLUTION SCHEMES FOR CONSERVATION LAWS AND CONVECTION-DIFFUSION EQUATIONS WITH VARYING TIME AND SPACE GRIDS

This paper presents a class of high resolution local time step schemes for nonlinear hyperbolic conservation laws and the closely related convection-diffusion equations, by projecting the solution increments of the underlying partial differential equations （PDE） at each local time step. The main advantages are that they are of good consistency, and it is convenient to implement them. The schemes are L^∞ stable, satisfy a cell entropy inequality, and may be extended to the initial boundary value problem of general unsteady PDEs with higher-order spatial derivatives. The high resolution schemes are given by combining the reconstruction technique with a second order TVD Runge-Kutta scheme or a Lax-Wendroff type method, respectively. The schemes are used to solve a linear convection-diffusion equation, the nonlinear inviscid Burgers＇ equation, the one- and two-dimensional compressible Euler equations, and the two-dimensional incompressible Navier-Stokes equations. The numerical results show that the schemes are of higher-order accuracy, and efficient in saving computational cost, especially, for the case of combining the present schemes with the adaptive mesh method [15]. The correct locations of the slow moving or stronger discontinuities are also obtained, although the schemes are slightly nonconservative.     

CONVERGENCE OF AN EXPLICIT UPWIND FINITE ELEMENT METHOD TO MULTI-DIMENSIONAL CONSERVATION LAWS

1. IntroductionThe convergence problem for the numerical schemes to one dimensional cons~inn lawshas been edensively studied. By tensor product one dimensional schemes can be appliedto multi-dimensional equations. However the convergence of many of those schemes is stillchanknown even if it is true for one dimensional cases. Besides, for those Phys.ical domains withcomplicated geometry unstructured grids are more fiekible. In recent yeaes the convergellceproblem for unstructured grids has cal…     

求解带刚性源项标量双曲型守恒律方程的保有界WCNS格式

带刚性源项的双曲守恒律方程是很多物理问题,特别是化学反应流的数学模型.本文考虑带刚性源项的标量双曲型守恒律方程,通过时空分离的方式,发展了一类保有界的WCNS格式.对于空间离散,我们将参数化的通量限制器推广到WCNS框架,使得方程对流项离散后满足极值原理.对于时间离散,我们将半离散的WCNS改写成指数形式,采用三阶修正...     

ON THE CENTRAL RELAXING SCHEMES I:SINGLE CONSERVATION LAWS

1. IntroductionIn [6], Jin and adn constructed a class of uPWind relaxing schemes for nonlinearconservation lawswith initial data u(0, x) ~ "o(x), x ~ (xl, ...t -cd), by using the idea of the local relaxation approximation [2,3,6,10].The relaxing scheme is obtained in the following way: A linear hyperbolic systemwith a stiff source term is first constructed to approximate the original equation (1.1)with a small dissipative correction. Then this linear hyperbolic system is solved easilyby und…     

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…     

COMPACT DIFFERENCE SCHEMES FOR THE DIFFUSION AND SCHRODINGER EQUATIONS. APPROXIMATION,STABILITY, CONVERGENCE,EFFECTIVENESS, MONOTONY

Various compact difference schemes （both old and new, explicit and implicit, one-level and two-level）, which approximate the diffusion equation and SchrSdinger equation with periodical boundary conditions are constructed by means of the general approach. The results of numerical experiments for various initial data and right hand side are presented. We evaluate the real order of their convergence, as well as their stability, effectiveness, and various kinds of monotony. The optimal Courant number depends on the number of grid knots and on the smoothness of solutions. The competition of various schemes should be organized for the fixed number of arithmetic operations, which are necessary for numerical integration of a given Cauchy problem. This approach to the construction of compact schemes can be developed for numerical solution of various problems of mathematical physics.     

依赖时间的Lagrange乘子区域分解法的显式格式

１．引言 近年来,一类新的区域分解法－非匹配网格区域分解法,日益引起人们的广泛兴趣．这类区域分解法的特点是：相邻子区域在公共边（或面）上的结点可以不重合,从而可方便地处理匹配网格区域分解法难以处理的问题：变动网格问题（例如石油勘探中的地层错动问题）和最优网格设计问题（即根据解的性质和实际问题的要求在不同子区域上采用不同的单元类型,不同的网格尺寸和不同阶的逼近多项式）． 在这类区域分解的算法设计中面临着两个困难：界面上非协调性的处理（与通常的协调元不同）和界面上积分的有效计算．现有算法中较引人注目的…     

结合气体分子动力学方法的RKDG有限元格式求解一维Euler可压方程组

In this paper, two numerical methods are developed for solving one-dimensionl compressible ELder equations by the RKDG finite element method.The schemesare obtained based on an important relation between the Boltzmann equation andthe ELder equations.The schemes have the TVD-like property under the uniform meshes.Several numerical results also present the performance of the schemes.     

数值求解Euler方程的UCGVC差分格式的注记

１．引言 近年来高精度差分格式的研究引起国内外的普遍重视,目的是更准确地模拟复杂流场的流动．众所周知,传统的二阶ＴＶＤ类格式虽然能较好地捕捉激波,但却存在局部极值点降阶的问题,而且由于一些格式的数值粘性过大,当用该格式计算粘性流特别是高雷诺数问题时,格式本身的数值粘性可能掩盖了流场的物理粘性,从而降低了格式对边界层的分辨率,因而无法正确计算热流值。文献［３］指出,采用高精度格式可适当放松对网格雷诺数的要求,因此发展三阶或三阶以上的格式是需要的。近年来,人们已经发展了一些无伪振荡的高阶格式,如ＥＮ…     

三级四阶显式辛R-K-N格式的完全构造

s级p阶辛Runge-Kutta-Nystr\"om(R-K-N)方法的一种充要条件是用关于参数的非线性方程组来表示的,辛R-K-N格式的构造问题因而转化为该方程组的求解问题. 在一些特殊的限定条件下, 已有该方程组在s=3,p=4时的两组解,即得到了两个三级四阶显式辛格式. 对于s=3,p=4情形,基于吴方法,利用计算机代数系统Maple及软件包wsolve给出了对应的非线性方程组的全部解, 这样就构造了所有的三级四阶显式辛R-K-N格式, 并证明了三级四阶显式辛R-K-N方法所满足的条件方程有冗余. 数值实验结果显示出新的辛格式在一定的条件下有着较好的误差精度.     

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…     

ON A CELL ENTROPY INEQUALITY OF THE RELAXING SCHEMES FOR SCALAR CONSERVATION LAWS

1.IntroductionIn[1],JinandXinconstructedaclassofthelocalrelaxationapproximation--TheRelaxingSchemesforsystemsofnonlinearcollservationlawsorsinglenonlinearconservationlawson,dofi(u)~ ;ac~0,(1l)otMoxil=1zwithinitialdatau(0,x)~"o(x),x~(xl,...,-cd),byusingtheideaofthelocalrelaxationapproximation[1--4].Theschemeisobtainedinthefollowingway:firstalinearhyperbolicsystemwithastiffsourcetermisconstructedtoapproximatetheoriginalsystem(1.1)withasmalldissipativecorrection;Then,thenewlinearhyperbolicsyste…