谱变形的非线性双曲型方程

文[1]研究了一个非线性双曲型方程 u_(xt) [(uu_xu_t)/(1-u~2)]-u(1-u~2)=0.方程(*)可以用来描述沿类脂膜扩张波的传播,它所对应的特征值问题的谱是不变的,随着科技的发展,非均匀介质中的传播问题日益受到重视。类问题相应的特征值问题是非保谱的,因此,在[2]中研究了谱变形的特征值问题和发展方程.这本文研究谱变形的非线性双曲型方程     

高维广义BBM方程的Chebyshev拟谱方法

在非线性长波的研究中[1],提出并研究了BBM方程.由于这类方程在很多数学物理问题中出现,如双温热传导的冷却过程,液体在碎石中的渗流问题等,因而引起了人们的关注.这类方程的数值方法,已有许多工作,但主要是采用差分法和有限元法.[2]使用.Fourier谱方法讨论了一维广义BBM方程,我们在[3]中也用Fourier谱和拟谱方法讨论了高维广义BBM方程.然而对于非周期情况,Fourier谱方法无法使用.在     

解高维广义BBM方程的谱方法和拟谱方法

在非线性色散介质的长波研究中,Benjanin,Bona和Mahony等人提出并讨论了BBM方程。这类方程在许多数学物理问题中出现,如热力学中的双温热传导问题、在岩石裂缝中的渗流问题等,因而引起了人们的重视。之后,Goldstein,Avrin,郭柏灵等进一步研究了高维广义BBM方程。这类方程的数值分析很多,但主要是差分法和有限元法,如[9-10],[11]在一维情形下用谱方法和拟谱方法作了研究。本文讨论高维     

高维广义Kuramoto-Sivashinsky型方程Fourier拟谱方法的大时间误差估计

在很多物理问题中出现如下方程:Kuramoto在研究反应扩散系统耗散结构时导出了上述方程,Sivashinsky在模拟火焰传播时也得到了它.此外,它还出现在粘性层流和Navier-Stokes方程的分枝解中.在[5-8]中,作者研究了一维情形下周期初值问题的整体吸引子和分枝解;[9]提出了广义KS型方程;[10-14]中研究了它的光滑解的存在性和t→+∞时的渐近性     

Kolmogorov-Spieqel-Siveshinky方程大时间问题的Fourier拟谱逼近

该文讨论了Kolmogorov-Spieqel-Siveshinky方程的周期初值问题, 研究了半离散Fourier拟谱解的长时间行为, 证明了半离散系统的收敛性和整体吸引子的存在性. 构造了全离散的三层显式Fourier拟谱格式, 并证明了该格式的收敛性, 最后通过数值计算验证了格式的可信性. 数值结果表明: 该格式是长时间稳定并可取时间大步长. 作者模拟了方程的解在相空间的轨线, 得到了一些有意义的结论.     

直径为d的n阶树的谱半径

二分图的特征值在量子化学中有意义，因此研究其图论性质和其特征值间的关系是有背景的，设Pd＋1（[d＋2/2],n-d-1,Pd＋1（[d＋4/2],n-d-1）分别为路Pd＋1的第[d＋2/2]和第[d＋4/2]个顶点上接出n-d-1条悬挂边所得到的树，本文证明了：若把所有直径为d（d≥1）的n阶树按其最大特征值从大到小的顺序排列，则排在前两位的依次是Pd＋1（[d＋2/2],n-d-1,Pd＋1（[d＋4/2],n-d-1） 。     

带五次项的NLS方程及其谱逼近的整体吸引子的维数估计

通过给出一般发展方程和其近似方程解的整体吸引子的Hausdorff维数上界间的关系，继[1，2]的讨论，本文进一步得到了带五次项的NLS方程和半离散Fourier谱近似解的整体吸引子的Hausdorff维数的上界估计。     

非局部Kuramoto-Sivashinsky方程谱逼近的大时间性态

1 引言带非局部项非线性四阶Kuramoto-Sivashinsky方程(NK-S)方程是数学物理中的重要方程(见(1．1)),它描述了混合气体燃烧过程中火焰是如何随时间演变的。随着无穷维动力系统研究的深入,人们越来越关注方程解的大时间性态．在文[3]中作者研究了这类     

一类分块矩阵的谱包含域

给出了一类复分块矩阵的谱包含域，推广与改进了文[1～5]的相应结果．     

广义KdV-Burgers方程长时间性态的谱方法

1．引言随着无限维动力系统研究的发展和深入,人们对非线性发展方程长时间性态的研究越来越重视[1－6］,而这种研究在很大程度上依赖于数值计算的结果,因此,计算结果是否可靠,计算格式先得是否合适都是值得探讨和深入研究的问题17－10．广义KdV－Burg6rs方程是一类重要的非线性发展方程,在实际问题中也有着广泛的应用,因此,对它的研究即有理论价值也有实际意义．本文讨论如下的广义KdV－Burgers方程的周期初值问题其中a,q是已知实常数,且a＞0八。）,g（。）,h00是已知实函数．文［10］对上述问题构造了半离散的Fourier谱逼近…     

Regularized additive operator-difference schemes

The construction of additive operator-difference (splitting) schemes for the approximate solution Cauchy problem for the first-order evolutionary equation is considered. Unconditionally stable additive schemes are constructed on the basis of the Samarskii regularization principle for operator-difference schemes. In the case of arbitrary multicomponent splitting, these schemes belong to the class of additive full approximation schemes. Regularized additive operator-difference schemes for evolutionary problems are constructed without the assumption that the regularizing operator and the operator of the problem are commutable. Regularized additive schemes with double multiplicative perturbation of the additive terms of the problem’s operator are proposed. The possibility of using factorized multicomponent splitting schemes, which can be used for the approximate solution of steadystate problems (finite difference relaxation schemes) are discussed. Some possibilities of extending the proposed regularized additive schemes to other problems are considered.     

波动方程两种哈密顿型蛙跳格式

1.构造格式 考虑如下波动方程 u_(tt)=u_(xx) (1.1)的初边值问题,设其边界条件为周期的,即在此条件下,解具有周期性.(1.1)有二种namilton形式.一种是经典形式:     

Nonstandard finite difference schemes for a class of generalized convection–diffusion–reaction equations

In this work, a class of nonstandard finite difference (NSFD) schemes are proposed to approximate the solutions of a class of generalized convection–diffusion–reaction equations. First, in the case of no diffusion, two exact finite difference schemes are presented using the method of characteristics. Based on these two exact schemes, a class of exact schemes are presented by introducing a parameter α. Second, since the forms of these exact schemes are so complicated that they are not convenient to use, a class of NSFD schemes are derived from the exact schemes using numerical approximations. It follows that, under certain conditions about denominator function of time‐step sizes, these NSFD schemes are elementary stable and the solutions are positive and bounded. Third, by means of the Mickens' technique of subequations, a new class of implicit NSFD schemes are constructed for the full convection–diffusion–reaction equations. It is shown that, under certain parameters set, these NSFD schemes are capable of preserving the non‐negativity and boundedness of the analytical solutions. Finally, some numerical simulations are provided to verify the validity of our analytical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1288–1309, 2015     

Construction of splitting schemes based on transition operator approximation

The stability analysis of approximate solutions to unsteady problems for partial differential equations is usually based on the use of the canonical form of operator-difference schemes. Another possibility widely used in the analysis of methods for solving Cauchy problems for systems of ordinary differential equations is associated with the estimation of the norm of the transition operator from the current time level to a new one. The stability of operator-difference schemes for a first-order model operator-differential equation is discussed. Primary attention is given to the construction of additive schemes (splitting schemes) based on approximations of the transition operator. Specifically, classical factorized schemes, componentwise splitting schemes, and regularized operator-difference schemes are related to the use of a certain multiplicative transition operator. Additive averaged operator-difference schemes are based on an additive representation of the transition operator. The construction of second-order splitting schemes in time is discussed. Inhomogeneous additive operator-difference schemes are constructed in which various types of transition operators are used for individual splitting operators.     

Commutative Association Schemes Whose Symmetrizations Have Two Classes

If a symmetric association scheme of class two is realized as the symmetrization of a commutative association scheme, then it either admits a unique symmetrizable fission scheme of class three or four, or admits three fission schemes, two of which are class three and one is of class four. We investigate the classification problem for symmetrizable (commutative) association schemes of two-class symmetric association schemes. In particular, we give a classification of association schemes whose symmetrizations are obtained from completely multipartite strongly regular graphs in the notion of wreath product of two schemes. Also the cyclotomic schemes associated to Paley graphs and their symmetrizable fission schemes are discussed in terms of their character tables.     

一类交错网格的Gauss型格式

本文在交错网格的情况下 ,利用 Gauss型求积公式构造了一类不需解 Riemann问题的求解一维单个双曲守恒律的二阶显式 Gauss型差分格式 ,证明了该格式在CFL条件限制下为 TVD格式 ,并证明了这类格式的收敛性 ,然后将格式推广到方程组的情形 .由于在交错网格的情况下构造的这类差分格式 ,不需要求解 Riemann问题 ,因此这类格式与诸如 Harten等的 TVD格式相比具有如下优点 :由于不需要完整的特征向量系 ,因此可用于求解弱双曲方程组 ,计算更快、编程更加简便等 .     

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89-112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge-Kutta methods, Appl. Numer. Math. 58 (2008) 1675-1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed.A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.     

Classification of association schemes with 18 and 19 vertices

The isomorphism classes of association schemes with 18 and 19 vertices are classified. We obtain 95 isomorphism classes of association schemes with 18 vertices and denote the representatives of the isomorphism classes as subschemes of 7 association schemes. We obtain 7 isomorphism classes of association schemes with 19 vertices and six of them are cyclotomic schemes.     

Minimal dissipation hybrid bicompact schemes for hyperbolic equations

New monotonicity-preserving hybrid schemes are proposed for multidimensional hyperbolic equations. They are convex combinations of high-order accurate central bicompact schemes and upwind schemes of first-order accuracy in time and space. The weighting coefficients in these combinations depend on the local difference between the solutions produced by the high- and low-order accurate schemes at the current space-time point. The bicompact schemes are third-order accurate in time, while having the fourth order of accuracy and the first difference order in space. At every time level, they can be solved by marching in each spatial variable without using spatial splitting. The upwind schemes have minimal dissipation among all monotone schemes constructed on a minimum space-time stencil. The hybrid schemes constructed has been successfully tested as applied to a number of two-dimensional gas dynamics benchmark problems.     

Optimal discrete and continuous mono‐implicit Runge–Kutta schemes for BVODEs

Recent investigations of discretization schemes for the efficient numerical solution of boundary value ordinary differential equations (BVODEs) have focused on a subclass of the well‐known implicit Runge–Kutta (RK) schemes, called mono‐implicit RK (MIRK) schemes, which have been employed in two software packages for the numerical solution of BVODEs, called TWPBVP and MIRKDC. The latter package also employs continuous MIRK (CMIRK) schemes to provide C ¹ continuous approximate solutions. The particular schemes implemented in these codes come, in general, from multi‐parameter families and, in some cases, do not represent optimal choices from these families. In this paper, several optimization criteria are identified and applied in the derivation of optimal MIRK and CMIRK schemes for orders 1–6. In some cases the schemes obtained result from the analysis of existent multi‐parameter families; in other cases new families are derived from which specific optimal schemes are then obtained. New MIRK and CMIRK schemes are presented which are superior to those currently available. Numerical examples are provided to demonstrate the practical improvements that can be obtained by employing the optimal schemes. This revised version was published online in June 2006 with corrections to the Cover Date.