Alternating Direction Finite Volume Element Methods for Three-Dimensional Parabolic Equations

<正>This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes,one is analogous to Douglas finite difference scheme with second-order splitting error,the other two schemes have third-order splitting error,and the last one is an extended LOD scheme.The L~2 norm and H~1 semi-norm error estimates are obtained for the first scheme and second one,respectively.Finally,two numerical examples are provided to illustrate the efficiency and accuracy of the methods.     

A FRACTIONAL NONLINEAR EVOLUTIONARY DELAY SYSTEM DRIVEN BY A HEMI-VARIATIONAL INEQUALITY IN BANACH SPACES

This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space.Utilizing the KKM theorem,a result concerned with the upper semicontinuity and measurability of the solution set of a hemivariational inequality is established.By using a fixed point theorem for a condensing setvalued map,the nonemptiness and compactness of the set of mild solutions are also obtained for such a system under mild conditions.Finally,an example is presented to illustrate our main results.     

LIMIT CYCLES FOR A CLASS OF NONPOLYNOMIAL PLANAR VECTOR FIELDS(II)

In this paper, the problem of limit cycles for a class of nonpolynomial planar vector felds is investigated. First, based on Liapunov method theory, we obtain some sufcient conditions for determining the origin as the critical point of such nonpolynomial planar vector felds to be the focus or center. Then, using Dulac criterion, we establish some sufcient conditions for the nonexistence of limit cycles of this nonpolynomial planar vector felds. And then, according to Hopf bifurcation theory, we analyze some sufcient conditions for bifurcating limit cycles from the origin. Finally, by transforming the nonpolynomial planar vector felds into the generalized Li′enard planar vector felds, we discuss the existence, uniqueness and stability of limit cycles for the former and latter planar vector felds. Some examples are also given to illustrate the efectiveness of our theoretical results.     

A Finite Difference Scheme for Blow-Up Solutions of Nonlinear Wave Equations

<正>We consider a finite difference scheme for a nonlinear wave equation,whose solutions may lose their smoothness in finite time,i.e.,blow up in finite time.In order to numerically reproduce blow-up solutions,we propose a rule for a time-stepping, which is a variant of what was successfully used in the case of nonlinear parabolic equations.A numerical blow-up time is defined and is proved to converge,under a certain hypothesis,to the real blow-up time as the grid size tends to zero.     

Reflected SPDEs Driven by Fractional Noises

In this paper,we study the stochastic partial differential equation with two reflecting smooth walls h^1 and h^2,driven by a fractional noise,which is fractional in time and white in space.The large deviation principle for the law of the solution to this equation,will be established through developing a classical method.Furthermore,we obtain the H?lder continuity of the solution.     

Maximum Likelihood Estimator of the Location Parameter under Moving Extremes Ranked Set Sampling Design

Cost effective sampling design is a problem of major concern in some experiments especially when the measurement of the characteristic of interest is costly or painful or time consuming.In the current paper,a modification of ranked set sampling(RSS)called moving extremes RSS(MERSS)is considered for the estimation of the location parameter for location family.A maximum likelihood estimator(MLE)of the location parameter for this family is studied and its properties are obtained.We prove that the MLE is an equivariant estimator under location transformation.In order to give more insight into the performance of MERSS with respect to(w.r.t.)simple random sampling(SRS),the asymptotic efficiency of the MLE using MERSS w.r.t.that using SRS is computed for some usual location distributions.The relative results show that the MLE using MERSS can be real competitors to the MLE using SRS.     

立体几何中的活跃分子——直角四面体

三条侧棱两两相互垂直的四面体是一种特殊的四面体,我们称之为直角四面体,它具有以下性质:(1)任何一条侧棱垂直另两个侧棱构成的平面;(2)三个侧面两两垂直;(3)顶点在底面上的射影是底面三角形的垂心等,立体几何中很重要的概念和定理.都能从这个直角四正面体中衍生,因此深入研究直角四面体,对于把握空间图形中直线和平面的关系,尤为重要.下面利用直角四面体的性质简解两道商考题.……     

Time variant multi-objective linear fractional interval-valued transportation problem

This paper studies a time-variant multi-objective linear fractional transportation problem. In reality, transported goods should reach in destinations within a specific time. Considering the importance of time, a time-variant multi-objective linear fractional transportation problem is formulated here. We take into account the parameters as cost, supply and demand are interval valued that involved in the proposed model, so we treat the model as a multi-objective linear fractional interval transportation problem. To solve the formulated model, we first convert it into a deterministic form using a new transformation technique and then apply fuzzy programming to solve it. The applicability of our proposed method is shown by considering two numerical examples. At last, conclusions and future research directions regarding our study is included.     

Long-Time Oscillatory Energy Conservation of Total Energy-Preserving Methods for Highly Oscillatory Hamiltonian Systems

For an integrator when applied to a highly oscillatory system,the near conservation of the oscillatory energy over long times is an important aspect.In this paper,we study the long-time near conservation of oscillatory energy for the adapted average vector field(AAVF)method when applied to highly oscillatory Hamiltonian systems.This AAVF method is an extension of the average vector field method and preserves the total energy of highly oscillatory Hamiltonian systems exactly.This paper is devoted to analysing another important property of AAVF method,i.e.,the near conservation of its oscillatory energy in a long term.The long-time oscillatory energy conservation is obtained via constructing a modulated Fourier expansion of the AAVF method and deriving an almost invariant of the expansion.A similar result of the method in the multi-frequency case is also presented in this paper.     

On Global and Local Properties of the Trajectories of Gaussian Random Fields——A Look Through the Set of Limit Points

This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen's functional laws of the iterated logarithm at zero and infinity respectively.The sets of limit points of those Gaussian random fields are obtained.The main results are applied to fractional Riesz-Bessel processes and the sets of limit points of this field are obtained.     

一类多维非线性反应扩散方程有限差分格式的稳定性问题(英文)

本文研究了一类多维线性反应扩散方程差分格式的稳定性.利用量未知元方法,建立了具有增量未知元的有限差分格式;然后利用非线性Galerkin方法,得到该差分格式的稳定性条件.通过对该格式的稳定性分析,说明和经典的差分格式的稳定性相比较,带有增量未知元的有限差分格式的稳定性得到了提高.     

Nonlinear Galerkin method in the finite difference case and wavelet-like incremental unknowns

Summary The IMG algorithm (Inertial Manifold-Multigrid algorithm) which uses the first-order incremental unknowns was introduced in [20]. The IMG algorithm is aimed at numerically implementing inertial manifolds (see e.g. [19]) when finite difference discretizations are used. For that purpose it is necessary to decompose the unknown function into its long wavelength and its short wavelength components; (first-order) Incremental Unknowns (IU) were proposed in [20] as a means to realize this decomposition. Our aim in the present article is to propose and study other forms of incremental unknowns, in particular the Wavelet-like Incremental Unknowns (WIU), so-called because of their oscillatory nature.In this report, we first extend the general convergence results in [20] by proving them under slightly weaker conditions. We then present three sets of incremental unknowns (i.e. the first-order as in [20], the second-order and wavelet-like incremental unknowns). We show that these incremental unknown can be used to construct convergent IMG algorithms. Special stress is put on the wavelet-like incremental unknowns since this set of unknowns has theL ² orthogonality property between different levels of unknowns and this should make them particularly appropriate for the approximation of evolution equations by inertial algorithms.     

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.     

A modified Crank-Nicolson scheme with incremental unknowns for convection dominated diffusion equations

A modified Crank-Nicolson scheme based on one-sided difference approximations is proposed for solving time-dependent convection dominated diffusion equations in two-dimensional space. The modified scheme is consistent and unconditionally stable. A priori L² error estimate for the fully discrete modified scheme is derived. With the use of the incremental unknowns preconditioner at each time step, a comparison among several classical numerical schemes has been made and numerical results confirm stability and efficiency of the modified Crank-Nicolson scheme.     

一类非线性Schr(o)dinger方程的守恒差分法与Fourier谱方法

考察了一类带导数项的非线性Schrodinger方程的周期边值问题，提出了一种守恒的差分格式，在空间方向上采用Fourier谱方法，证明了格式的稳定性和收敛性．数值试验得到了与理论分析一致的结果．     

Finite Volume Multilevel Approximation of the Shallow Water Equations

The authors consider a simple transport equation in one-dimensional space and the linearized shallow water equations in two-dimensional space, and describe and implement a multilevel finite-volume discretization in the context of the utilization of the incremental unknowns. The numerical stability of the method is proved in both cases.     

一类非线性反应-扩散方程有限差分格式的稳定性研究

In the article,the fully discrete finite difference scheme for a type of nonlinear reaction-diffusion equation is established.Then the new function space is introduced and the stability problem for the finite difference scheme is discussed by means of variational approximation method in this function space.The approach used is of a simple characteristic in gaining the stability condition of the scheme.     

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.     

Stability of a colocated finite volume scheme for the Navier‐Stokes equations

The aim of this article is to describe a colocated finite volume approximation of the incompressible Navier‐Stokes equation and study its stability. One of the advantages of colocated finite volume space discretizations over staggered space discretizations is that all the variables share the same location; hence, the possibility to more easily use complex geometries and hierarchical decompositions of the unknowns. The time discretization used in the scheme studied here is a projection method. First, we give the full discretization of the incompressible Navier‐Stokes equations, then, we state the stability result and prove it following the methods of Marion and Temam. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Incremental unknowns for solving partial differential equations

Summary Incremental unknowns have been proposed in [T] as a method to approximate fractal attractors by using finite difference approximations of evolution equations. In the case of linear elliptic problems, the utilization of incremental unknown methods provides a new way for solving such problems using several levels of discretization; the method is similar but different from the classical multigrid method.In this article we describe the application of incremental unknowns for solving Laplace equations in dimensions one and two. We provide theoretical results concerning two-level approximations and we report on numerical tests done with multi-level approximations.