Taylor expansion method for nonlinear evolution equations

Introduction Thestudyofnonlinearevolutionequationsisafascinatingproblemwhichisattheveryheart oftheunderstandingofmanyimportantproblemsinthenaturalsciences[1,2].Thenonlinear evolutionequationsandtheirnumericalapproximationareveryimportantintheareasof theoreticalmathematicsandcomputationalmathematics.Aninterestingfeatureofthe approximationtheoryofthenonlinearevolutionequationsistheapplicationsofthefunctional analyticmethodstothenumericalapproximationofthenonlinearevolutionequations. Thispaperist…     

ON THE NUMERICAL SOLUTION OF QUASILINEAR WAVE EQUATION WITH STRONG DISSIPATIVE TERM

The numerical solution for a type of quasilinear wave equation is studied. The three-level difference scheme for quasi-linear waver equation with strong dissipative term is constructed and the convergence is proved. The error of the difference solution is estimated. The theoretical results are controlled on a numerical example.     

TWO-GRID METHOD FOR CHARACTERISTICS FINITE-ELEMENT SOLUTION OF 2D NONLINEAR CONVECTION-DOMINATED DIFFUSION PROBLEM

Introduction Convection diffusionequationgovernssuchphenomenaastheflowofheatwithina movingfluid,thetransportofdissolvednutrientsorcontaminantswithinthegroundwater,andthetransportofasurfactantortracerwithinanincompressibleoilinapetroleum reservoir.Weconsid…     

THE THREE-DIMENSIONAL UNSTEADY BOUNDARY LAYER OVER AN IMPULSIVELY STARTED ROTATING DISK

The unsteady boundary layer over an impulsively started rotating disk isstudied,a complete solution describing the smooth transition from vortex diffusionat ωt=0 to Kármán’s steady solution is obtained by series expansion and itsnumerical continuation. The angle of body streamlines,together withexperimental values, are given as the function of time t as well as the momentcoefficient C_M and on-coming velocity w(∞).     

A WAVE EQUATION MODEL TO SOLVE THE MULTIDIMENSIONAL TRANSPORT EQUATION

The wave equation model, originally developed to solve the advection–diffusion equation, is extended to the multidimensional transport equation in which the advection velocities vary in space and time. The size of the advection term with respect to the diffusion term is arbitrary. An operator-splitting method is adopted to solve the transport equation. The advection and diffusion equations are solved separate ly at each time step. During the advection phase the advection equation is solved using the wave equation model. Consistency of the first-order advection equation and the second-order wave equation is established. A finite element method with mass lumping is employed to calculate the three-dimensional advection of both a Gaussian cylinder and sphere in both translational and rotational flow fields. The numerical solutions are accurate in comparison with the exact solutions. The numerical results indicate that (i) the wave equation model introduces minimal numerical oscillation, (ii) mass lumping reduces the computational costs and does not significantly degrade the numerical solutions and (iii) the solution accuracy is relatively independent of the Courant number provided that a stability constraint is satisfied. © 1997 by John Wiley & Sons, Ltd.     

用差分法与超松弛迭代法求高维平稳FPK方程的解

用不同精度的差分格式将高维平稳FPK方程离散化为线性代数方程组,然后用超松弛迭代法求解该线性代数方程组得到平稳FPK方程的近似解。讨论了不同的差分格式、网格密度及超松弛因子对解精度及收敛速度的影响,并与其他方法的计算精度进行比较,提出用多重网格算法提高计算效率。研究了典型的二维及四维随机系统的稳态响应,算例表明,该算法具有简洁、节省存储量且精度高的特点,是求解高维平稳FPK方程解的有效算法。     

A 1/3 Pure Subharmonic Solution and Transient Process for the Duffing's Equation

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Falkner-Skan方程的近似解析解

研究了粘性流体绕流楔型物体的Falkner-Skan边界层方程求解问题.利用Adomian拆分方法,通过引入Crocco变量变换将无穷区间的边界值问题转为初值问题并利用Padé逼近技巧确定初值,给出了一种有效的解析分解方法.进一步,本文设计了一种数值解法,将本文得到的近似解析解及数值结果与早期研究者Hartree等人的结果进行了比较,证明了本文提出的解法的有效性和可靠性.     

一类非线性奇异积分方程及其数值方法研究

探讨了一类非线性奇异积分方程的数理性质以及在颗粒雷诺数Rep＜1时此类方程解的存在条件，然后详细研究了该方程的数值计算方法并构造称之为P（EC）^k多步法的差分格式，分析了该格式的收敛性和代数精度，得到时域离散步长的约束关系。运用该格式计算了静止流场和均匀振荡流中球形小颗粒的非恒定运动，将计算结果与其解析解及有关实验数据的比较表明，上述数值方法具有良好的计算精度。     

SAINT-VENANT’S TORSION PROBLEM FOR A COMPOSITE CIRCULAR CYLINDER WITH ANINTERNAL EDGE CRACK

In this paper the writer uses Muskhelishvili single-layer potential function solutionand single crack solution for the torsion problem of a circular cylinder to discuss thetorsion problem of a composite cylinder with an internal crack,and the problem isreduced to a set of mixed-type integral equation with generalized Cauchy-kernel.Then,by using the integration formula of Gauss-Jacobi.the numerical method isestablished and several numerical examples are calculated.The torsional rigidity andthe stress intensity factors are obtained.The results of these examples fit the resultsobtained by the previous papers better.     

弹性力学Hamilton方法广义解的适定性

首先引入了Hamilton体系中平面应力弹性力学问题正则方程的Galerkin变分方程,证明了Galerkin变分方程和目前文献中所用的Ritz变分方程的等价性,以及相应广义解的适定性,从而为目前的数值方法提供了理论基础。从证明过程中可以看到广义解实际上是Ritz变分泛函的一个鞍点。     

Numerical Simulation of Isotropic Turbulence Dynamics

The Karman-Howarth closed equation is used to describe the behavior of homogeneous isotropic turbulence. A numerical solution is obtained by the collocation-grid and finite-difference methods using moving grids. The predicted results are compared with experimental data on the decay of fully developed and weak turbulence. The numerical realization of the Loitsiansky-MilHonshtchikov asymptotic solution has been made.     

一类非线性周期系统响应的精细积分法

对于一类非线性周期／变系数微分方程,提出基于精细积分法的数值解法,处理非线性周期／变系数微分方程系统的响应问题,其积分策略是：采用精细积分格式处理常系数部分;采用线性插值格式处理非线性周期／变系数部分,既继承精细积分方程高度准确的特点,又保证足够的精度与较小的计算量。通过数值算例,与以往与用的微分方程直接数值积分法（如预估－校正哈明法）求得的解加以比较表明,对于给定的精度要求,精细积分法更经济有效     

High-order boundary conditions for the problems of Laplace equation in infinite region and their application

The high-order boundary conditions for the problems of Laplace equation in infiniteregion have been developed.The improvement in accuracy for numerical solution isachieved by imposing the high-order boundary conditions on the exterior boundary of areduced finite region in which the numerical method is used.So both the computing effortsand the required storage in computer are reduced.The numerical examples show that thelst-order boundary condition approaches to the exact boundary condition and it is clearlysuperior to the traditional boundary condition and the2nd-order boundary condition.     

广义Stokes方程的完全边界积分表示式及其在求解N-S方程中的应用

为了分解N-S方程组各变量相互偶合,本文采用Peaceman-Rachford算子分裂法,将时间相依的N-S方程组分解成不存在上述偶合特性的线性和非线性的子问题。线性子问题具有广义Stokes方程类型。本文采用多重互易法,即采用多阶拉普拉斯算子基本解逐步变换,将其解表示成完全边界积分形式,从而使问题的计算维数降低一维。广义Stokes方程的算例以及二维圆柱在剪切流中的Stokes绕流解,都表明多重互易算法具有高效特点,而且后者与文[3]解析解吻合得非常好。     

THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PROBLEM FOR SEMILINEAR PARABOLIC DIFFERENTIAL EQUATION

The numerical solution of a singularly perturbed problem for the semilinear parabolicdifferential equation with parabolic boundary layers is discussed.A nonlinear two-leveldifference scheme is constructed on the special non-uniform grids.The uniform convergenceof this scheme is proved and some numerical examples are given.     

Numerical solution of multiple hole problem by using boundary integral equation

This paper studies a numerical solution of multiple hole problem by using a boundary integral equation. The studied problem can be considered as a supposition of many single hole problems. After considering the interaction among holes, an algebraic equation is formulated, which is then solved by using an iteration technique. The hoop stress around holes can be finally determined. One numerical example is provided to check its accuracy.     

A solution to the hydrodynamic lubrication of a circular point contact sliding over a flat surface with cavitation

This letter presents an analytical solution to the hydrodynamic lubrication of a circular point contact sliding over a flat surface with cavitation. The solution is found by solving the Reynolds equation with Reynolds boundary condition for cavitation. The cavitation boundary is shown to be straight lines directed 108.4° against the sliding direction. The result is experimentally verified in the limit of large values of viscosity, sliding velocity and radius of a spherical ball. The solution raises questions about the coupling between cavitation and film rupture and can be used as an independent check on the validity of numerical solutions.     

The numerical solution of a singularly perturbed problem for quasilinear parabolic differential equation

We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform convergence in the sense of discrete L~2 norm is proved and numerical examples are presented.     

Exponential dichotomies of nonlinear discrete systems and its application to numerical analysis and computation

In this pepar we consider the upwind difference scheme of a kind of boundary value problems for nonlinear, second order, ordinary differential equations. Singular perturbation method is applied to construct the asymptotic approximation of the solution to the upwind difference equation. Using the theory of exponential dichotomies we show that the solution of an order-reduced equation is a good approximation of the solution to the upwind difference equation except near boundaries. We construct correctors which yield asymptotic approximations by adding them to the solution of the order-reduced equation. Finally, some numerical examples are illustrated.