一种简化的三阶精度加权ENO格式

91．引言从七十年代后期开始，对双曲型守恒律方程数值方法的研究以VanLeer构造出来的MUSCL格式［‘］为先导，出现了一些全新的高分辨率守恒型差分格式．特别是A．Harten［‘］提出了TVD（TotalVariationDiminising）格式的概念后，双曲型守恒律方程数值方法的研究取得了飞速的发展．因为TVD格式可以保持数值解的单调性，所以它可以有效地抑制间断附近数值解的振荡，这方面有重要代表性的工作是［2－4］．由于TVD格式必须保证数值解的总变差不增，所以使得TVD格式在光滑解的局部极值点处降价．为了克服TVD格式的这个弱点，便出…     

(J,J′)-无损因子分解中的若干问题研究

（Ｊ，Ｊ′）┐无损因子分解中的若干问题研究１）裘聿皇张本勇（中国科学院自动化研究所北京１０００８０）关键词链式散射描述，（Ｊ，Ｊ′）－无损因子分解，Ｈ∞控制问题．１）国家自然科学基金资助课题．收稿日期１９９５－１１－１７１引言在文［１］中利用链式散射...     

非整数阶微积分的滤波特性及数值算法

１．引言在工程计算中,时常遇到对实验观测数据求非整数阶微积分的数值计算问题,如精确变换中Ｈｉｌｂｅｒｔ变换的计算［１］,二维波动方程偏移［２］,分数维计算［３－５］等都涉及到了非整数阶微积分的概念与数值计算．从而使非整数阶微积分的数值计算成为不可忽视的问题．本文借助Ｆｏｕｒｉｅｒ变换,利用线性时不变系统滤波理论,从滤波角度探讨了微积分的滤波特性,讨论了微积分数值计算的快速傅氏变换（ＦＦＴ）算法及其适用条件,并将该算法推广到非整数阶微积分的数值计算,通过算例进一步说明了所述理论与算法的正确性．这对…     

关于非对称线性方程组的新迭代算法

５１．引言 二阶椭圆型非对称方程是一类重要的科学工程计算的数学模型,如对流扩散和油藏模拟方程等,有着广泛的实际应用背景．文献［２］和［３］基于原始微分方程及对应的离散问题提出了正定可对称化的新概念．基于这一概念及文山我们研究针对二维和三维二阶常系数非对称椭圆型方程数值模型的新选代算法,首先考虑下面的一维椭圆型问题：对区间［０,１］均匀剖分后得ｎ＋２节点,即ｘ;一ｉ·ｈ;ｉ＝０,··,,ｎ＋１,其中ｈ＝／（＋１）．如果用中心差分格式离散方程（１．１）,则在节点Ｘ;有如下差分方程： 一（１＋ｗＮ。;－…     

基于散射模型的Ｈ∞ 控制系统的稳定性*

本文利用（Ｊ，Ｊ＇）－无损矩阵的性质，研究了基于散射模型的Ｈ∞控制系统的闭环稳定性，获得了广义对象的散射模型为（Ｊ，Ｊ＇）－无损的Ｈ∞控制问题可解的充要条件。与文［１］中关于广义对象为一类内矩阵的Ｈ∞控制问题求解的结果相比，本文的结论对广义对象的限制要弱。     

欧拉方程的velicity表示和辛积分(Ⅱ)

１．引言这篇文章的目的是辨别辛方法得出的结果与龙格－库塔法相比是同样好或是更好,特别对长时间．文中的数值实验显示对于ｔ＝０．１,用ＩＭ格式,在ｔ＝６时将得到不均匀分布的点．２．计算的描述本文将显示用［４］中描述的近似获得的数值结果．结论写在末尾．对［４］中的哈密顿系统（２１）,（２２）,本文将对隐式中点格式（ＩＭ）和二级四阶高斯－勒让德龙格－库塔方法与标准四阶龙格－库塔方法作比较．在数值实验中用了三个不同的哈密顿函数：一个是［４］中哈密顿函数（１９）,其他的是通过省略ｋ＝ｊ项和双倍这项从［４］中…     

对流占优扩散问题的并行计算

１．引言 在刻画流体运动的某些物理现象,以及研究热的传导、粒子的扩散等问题时,都会归结到求解对流扩散方程．用有限差分方法求解该方程,若采用显式方法,计算格式简单,但它们都是条件稳定的,时间步长必须取得非常小;若采用隐式方法,方法是无条件稳定的,但要解代数方程组,求解比较困难．Ｄ．Ｊ．ＥＶＡＮＳ和Ａ．Ｒ．ＡＨＭＡＤ在文［２］中提出了用显式交替方向法求解定态椭圆型方程,对Ｌａｐｌａｃｅ方程做了数值实验．本文将这个方法推广到了时间依赖的问题,而且适用于对流占优扩散问题的求解．基于二阶迎风格式［１］;本…     

正则长波方程的一个新的差分方法

§１．引言正则长波（ＲＬＷ）方程在１９６６年由Ｐｅｒｅｇｒｉｎｅ［１］第一次提出,它描述波的运动有与ＫｄＶ方程相同的逼近界,并且它能够相当好地摸拟ＫｄＶ方程的所有应用,因此引起了人们的注意．文［２－６］讨论了它的数值方法,其中文［２］提出了一个两层的和一个三层的差分格式,它们分别具有一阶和二阶精度．文［４］考虑了一个守恒的两层差分格式．本文考虑以下ＲＬＷ方程的初边值问题这个问题具有以下能量守恒律方程（３）的单个孤波解为其中ａ和是任意常数．从（７）可以明显地看出当一ｘＬ和ｘＲ足够大时,初边值问题（…     

关于Ck-连续的保形插值样条函数的一点注记

针对有关Ｃｋ一连续的保形插值样条函数在确定边界导数的条件时可能出现多余的拐点,从而破坏了保形性［１,２］的问题,现提出一种修正方法． 我们采用与［２］相同的记号．对于区间［ａ, ｂ］的一个划分 ：ａ＝ｘ０＜ｘ１＜…＜ｘｎ＝ｂ,在每个节点ｘｉ处给定相应的型值ｙｉ,令 首先要指出,［２］之§５数值算例中的数据有误,β２＝－０．５－４＝－４．５而不是－５．５,与之对应的ｍ２＝ ４ ＋ ０．８９（－４．５）＝－０．００５．即使假定原始数据 β２＝一５．５,ｍ２＝－０．９是正确的,按他的原始方程所画的曲线上仍有两…     

非对称广义特征值问题的并行同伦-行列式算法

１．引言同伦算法是七十年代开始发展起来的求解非线性问题的数值方法．它的特点表现为通常是大范围收敛，容易实施并行计算．近二十年来，同伦算法的发展主要沿两条走线展开，即单纯形法和连续法．连续同伦算法的基本思路为：设Ｘ和Ｙ是Ｒ"中的非空子集，人ｘ：Ｘ－－＋Ｙ是光滑映射，如果对Ｖ（ｔ，ｘ）Ｅ［０，１］ｘＸ有Ｈ（ｔ，ｘ）一吨（ｘ）十（１一Ｏ八：）ＥＹ成立，则称光滑映射Ｈ：［０，１］ＸＸ＋Ｙ是ｆ和９之间的一个线性同伦．连续同伦算法主要是借助于同伦Ｈ的零点集Ｈ－'（０）从平凡映射ｇ在｛１｝ｘＲ"中的零点集｛１｝…     

Concepts and Application of Time-Limiters to High Resolution Schemes

A new class of implicit high-order non-oscillatory time integration schemes is introduced in a method-of-lines framework. These schemes can be used in conjunction with an appropriate spatial discretization scheme for the numerical solution of time dependent conservation equations. The main concept behind these schemes is that the order of accuracy in time is dropped locally in regions where the time evolution of the solution is not smooth. By doing this, an attempt is made at locally satisfying monotonicity conditions, while maintaining a high order of accuracy in most of the solution domain. When a linear high order time integration scheme is used along with a high order spatial discretization, enforcement of monotonicity imposes severe time-step restrictions. We propose to apply limiters to these time-integration schemes, thus making them non-linear. When these new schemes are used with high order spatial discretizations, solutions remain non-oscillatory for much larger time-steps as compared to linear time integration schemes. Numerical results obtained on scalar conservation equations and systems of conservation equations are highly promising.     

A posteriori error estimation and adaptive mesh refinement for combined thermal-stress finite element analysis

Local and global error estimators and an associated h-based adaptive mesh refinement schemes are proposed for coupled thermal-stress problems. The error estimators are based on the “flux smoothing” technique of Zienkiewicz and Zhu with important modifications to improve convergence performance and computational efficiency. Adaptive mesh refinement is based on the concept of adaptive accuracy criteria, previously presented by the authors for stress-based problems and extended here for coupled thermal-stress problems. Three methods of mesh refinement are presented and numerical results indicate that the proposed method is the most efficient in terms of number of adaptive mesh refinements required for convergence in both the thermal and stress solutions. Also, the proposed method required a smaller number of active degrees of freedom to obtain an accurate solution.     

Computations of steady and unsteady transport of pollutant in shallow water

We present new models for simulating the steady and unsteady transport of pollutant. Then the simple central-upwind schemes based on central weighted essentially non-oscillatory reconstructions are proposed in this paper for computing the one- and two-dimensional steady and unsteady models. Since the non-uniform width of the different local Riemann fans is calculated more accurately, the central-upwind schemes enjoy a much smaller numerical viscosity as well as the staggering between two neighboring sets of grids is avoided. Synchronously, due to the central-upwind schemes are combined with the fourth-order central weighted essentially non-oscillatory reconstructions, the schemes have the non-oscillatory behavior. The numerical results show the desired accuracy, high-resolution, and robustness of our methods.     

A new MHD code with adaptive mesh refinement and parallelization for astrophysics

A new code, named MAP, is written in FORTRAN language for magnetohydrodynamics (MHD) simulations with the adaptive mesh refinement (AMR) and Message Passing Interface (MPI) parallelization. There are several optional numerical schemes for computing the MHD part, namely, modified Mac Cormack Scheme (MMC), Lax–Friedrichs scheme (LF), and weighted essentially non-oscillatory (WENO) scheme. All of them are second-order, two-step, component-wise schemes for hyperbolic conservative equations. The total variation diminishing (TVD) limiters and approximate Riemann solvers are also equipped. A high resolution can be achieved by the hierarchical block-structured AMR mesh. We use the extended generalized Lagrange multiplier (EGLM) MHD equations to reduce the non-divergence free error produced by the scheme in the magnetic induction equation. The numerical algorithms for the non-ideal terms, e.g., the resistivity and the thermal conduction, are also equipped in the code. The details of the AMR and MPI algorithms are described in the paper.     

Adaptive Central-Upwind Schemes for Hamilton–Jacobi Equations with Nonconvex Hamiltonians

This paper is concerned with computing viscosity solutions of Hamilton–Jacobi equations using high-order Godunov-type projection-evolution methods. These schemes employ piecewise polynomial reconstructions, and it is a well-known fact that the use of more compressive limiters or higher-order polynomial pieces at the reconstruction step typically provides sharper resolution. We have observed, however, that in the case of nonconvex Hamiltonians, such reconstructions may lead to numerical approximations that converge to generalized solutions, different from the viscosity solution. In order to avoid this, we propose a simple adaptive strategy that allows to compute the unique viscosity solution with high resolution. The strategy is not tight to a particular numerical scheme. It is based on the idea that a more dissipative second-order reconstruction should be used near points where the Hamiltonian changes convexity (in order to guarantee convergence to the viscosity solution), while a higher order (more compressive) reconstruction may be used in the rest of the computational domain in order to provide a sharper resolution of the computed solution. We illustrate our adaptive strategy using a Godunov-type central-upwind scheme, the second-order generalized minmod and the fifth-order weighted essentially non-oscillatory (WENO) reconstruction. Our numerical examples demonstrate the robustness, reliability, and non-oscillatory nature of the proposed adaptive method.     

Two adaptive wavelet algorithms for non-linear parabolic partial differential equations

This paper is concerned with the construction of wavelet based adaptive algorithms for the numerical resolution of evolution equations. The adaptivity is applied into two complementary directions. The first direction shares the approaches involved in classical adaptive finite element methods and is related to a solution dependent definition of spaces of approximation. The second direction is related to the approximation of evolution operators that is made solution dependent following the philosophy of essentially non-oscillatory schemes. After the construction of the schemes, numerical tests are provided.     

High Order Fast Sweeping Methods for Static Hamilton–Jacobi Equations

We construct high order fast sweeping numerical methods for computing viscosity solutions of static Hamilton–Jacobi equations on rectangular grids. These methods combine high order weighted essentially non-oscillatory (WENO) approximations to derivatives, monotone numerical Hamiltonians and Gauss–Seidel iterations with alternating-direction sweepings. Based on well-developed first order sweeping methods, we design a novel approach to incorporate high order approximations to derivatives into numerical Hamiltonians such that the resulting numerical schemes are formally high order accurate and inherit the fast convergence from the alternating sweeping strategy. Extensive numerical examples verify efficiency, convergence and high order accuracy of the new methods.     

Fifth-Order Weighted Power-ENO Schemes for Hamilton-Jacobi Equations

We design a class of Weighted Power-ENO (Essentially Non-Oscillatory) schemes to approximate the viscosity solutions of Hamilton-Jacobi (HJ) equations. The essential idea of the Power-ENO scheme is to use a class of extended limiters to replace the minmod type limiters in the classical third-order ENO schemes so as to improve resolution near kinks where the solution has discontinuous gradients. Then a weighting strategy based on appropriate smoothness indicators lifts the scheme to be fifth-order accurate. In particular, numerical examples indicate that the Weighted Power_{3ENO5 works for general HJ equations while the Weighted Power_{\inftyENO5 works for non-linear convex HJ equations. Numerical experiments also demonstrate the accuracy and the robustness of the new schemes     

A Robust Reconstruction for Unstructured WENO Schemes

The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order numerical methods for hyperbolic partial differential equations (PDEs). While WENO schemes on structured meshes are quite mature, the development of finite volume WENO schemes on unstructured meshes is more difficult. A major difficulty is how to design a robust WENO reconstruction procedure to deal with distorted local mesh geometries or degenerate cases when the mesh quality varies for complex domain geometry. In this paper, we combine two different WENO reconstruction approaches to achieve a robust unstructured finite volume WENO reconstruction on complex mesh geometries. Numerical examples including both scalar and system cases are given to demonstrate stability and accuracy of the scheme.