A coupling difference scheme for the numerical solution of a singular perturbation problem

In this paper, we consider a singularly perturbed problem without turning points. On a special diseretization mesh, a coupling difference scheme, resulting from central difference scheme and Abrahamsson-Keller-Kreiss box scheme, is proposed and the second order convergence, uniform in the small parameter, is proved. Finally, numerical resulls are provided.     

An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection-diffusion equations

In this article, we have developed an overlapping Schwarz method for a weakly coupled system of convection-diffusion equations. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region, we use the central finite difference scheme on a uniform mesh, whereas on the nonlayer region, we use the mid-point difference scheme on a uniform mesh. It is shown that the numerical approximations converge in the maximum norm to the exact solution. We have proved that, when appropriate subdomains are used, the method produces almost second-order convergence. Furthermore, it is shown that two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantage of this method used with the proposed scheme is that it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.     

On the accuracy of the multiple scales method for non-linear vibrations of doubly curved shallow shells

Non-linear free and forced vibrations of doubly curved isotropic shallow shells are investigated via multi-modal Galerkin discretization and the method of multiple scales. Donnell’s non-linear shallow shell theory is used and it is assumed that the shell is simply supported with movable edges. By deriving two different forms of the stress function, the equations of motion are reduced to a system of infinite non-linear ordinary differential equations with quadratic and cubic non-linearities. A quadratic relation between the excitation and the fundamental frequency is considered and it is shown that, although in case of hardening non-linearities the results resemble those found via numerical integration or continuation softwares, in case of softening non-linearity the solution breaks down as the amplitude becomes larger than the thickness. Results reveal that, expressing the relation between the excitation and fundamental frequency in this form, which was considered by many researchers as a useful tool in analyzing strong non-linear oscillators, yields in spurious results when the non-linearity becomes of softening type.     

Solving systems of nonlinear difference equations by the multiple scales perturbation method

In this paper, we apply an improved version of the multiple scales perturbation method to a system of weakly nonlinear, regularly perturbed ordinary difference equations. Such systems arise as a result of the discretization of a system of nonlinear differential equations, or as a result in the stability analysis of nonlinear oscillations. In our procedure, asymptotic approximations of the solutions of the difference equations will be constructed which are valid on long iteration scales.     

A second order uniform difference scheme for a singularly perturbed turning point problem

We construct a positive type difference scheme.for a singularly perturbed boundary value problem with a turning point. It's proved that this scheme is the second order convergence, uniformly in ε, to the solution of the singularly perturbed B.V.P. Numerical examples are provided.     

Singular perturbation of boundary value problem of systems for quasilinear ordinary differential equations

In this paper,we study the singular perturbation of boundary value problem of systemsfor quasilinear ordinary differential equations:x′=f(t,x,y,ε),εy″=g(t,x,y,ε)y′ h(t,x,y,ε),x(0,ε)=A(ε),y(0,ε)=Bε,y(1,ε)=C(ε)where xf.y,h,A,B and C belong to R″and a is a diagonal matrix.Under the appropriateassumptions,using the technique of diagonalization and the theory of differentialinequalities we obtain the existence of solution and its componentwise uniformly validasymptotic estimation.     

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.     

Sloshing simulation of standing wave with time-independent finite difference method for Euler equations

The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional(2D) tank are studied.The irregular tank is mapped onto a fixed square domain through proper mapping functions. A staggered mesh system is employed in a 2D tank to calculate the elevation of the transient fluid.A time-independent finite difference method,which is developed by Bang-fuh Chen,is used to solve the Euler equations for incompressible and inviscid fluids.The numer...     

基于精细积分技术的非线性动力学方程的同伦摄动法

将精细积分技术（PIM）和同伦摄动方法（HPM）相结合,给出了一种求解非线性动力学方程的新的渐近数值方法。采用精细积分法求解非线性问题时,需要将非线性项对时间参数按Taylor级数展开,在展开项少时,计算精度对时间步长敏感;随着展开项的增加,计算格式会变得越来越复杂。采用同伦摄动法,则具有相对筒单的计算格式,但计算精度较差,应用范围也限于低维非线性微分方程。将这两种方法相结合得到的新的渐近数值方法则同时具备了两者的优点,既使同伦摄动方法的应用范围推广到高维非线性动力学方程的求解,又使精细积分方法在求解非线性问题时具有较简单的计算格式。数值算例表明,该方法具有较高的数值精度和计算效率。     

Comparison of multiple scales and KBM methods for strongly nonlinear oscillators with slowly varying parameters

Two versions of multiple scales and Krylov–Bogoliubov–Mitropolsky (KBM) methods are extended to obtain the asymptotic solutions of a class of strongly nonlinear oscillators with slowly varying parameters. A comparison of the two methods is made to show that the multiple scales method is equivalent to the KBM method for the first order approximation. An example of pendulum with slowly varying length is given to illustrate the comparison of the two methods.     

Precise integration method for solving singular perturbation problems

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.     

稀薄流非线性模型方程离散速度坐标法有限差分解

从一般非线性Bo ltzm ann方程出发,发展并实现了一套适于大范围K nudsen数稀薄流问题数值模拟的统一算法。采用BGK模型和Shakov模型近似碰撞项,进而引入两个二速度无量纲简化分布函数,通过关于分子速度第三分量取矩积分,将三速度单一模型方程变换为二速度微分方程组。基于G auss-H erm ite积分公式和正交多项式G auss积分公式,借助离散速度坐标法消除简化模型方程对分子速度空间的连续依赖性,从相空间到物理空间得到一组带源项双曲守恒离散方程,并给出其显式和隐式二阶迎风TVD有限差分解。以二维圆柱A r气体超声速绕流算例,验证了数值算法的有效性,比较分析了漫反射和镜面反射两种气体分子壁面反射模型的计算结果。     

悬索主共振响应的多尺度解与同伦分析解

利用哈密顿变分原理,引入拟静态假设,建立了悬索面内非线性运动方程,并采用Galerkin方法对其进行离散。接着运用多尺度法和同伦分析法得到了悬索前两阶模态主共振响应的近似解。为验证这两种分析方法的适用性,同时采用龙格-库塔法对方程直接进行了数值积分。数值计算结果表明,随着悬索垂跨比以及振幅的增加,由多尺度法与同伦分析法得到的幅频响应曲线存在明显的定性与定量的差别,而同伦分析法结果与数值法的结果更加接近。最后比较了两种分析方法得到的位移场与索力时程响应曲线。     

A perturbation boundary integral equation method for analysis of interaction between foundation and nonlinear rock and soil medium

A scheme is developed for analysing the interaction between a foundation and a nonlinear rock and soil medium, in which the foundation is considered as a linear elastic body and a typical boundary integral equation method (BIEM) is employed. On the basis of taking the nonlinear properties of the medium into account, a perturbation BIEM is developed. The fundamental equations for the nonlinear coupling analysis are formulated, and typical problems are solved and discussed by the present method.     

Finite difference streamline diffusion method using nonconforming space for incompressible time-dependent Navier-Stokes equations

This paper proposes a new nonconforming finite difference streamline diffusion method to solve incompressible time-dependent Navier-Stokes equations with a high Reynolds number. The backwards difference in time and the Crouzeix-Raviart （CR） element combined with the P0 element in space are used. The result shows that this scheme has good stabilities and error estimates independent of the viscosity coefficient.     

A UNIFORMLY CONVERGENT DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF A HIGH ORDER ELLIPTIC DIFFERENTIAL EQUATION

ＡＵＮＩＦＯＲＭＬＹＣＯＮＶＥＲＧＥＮＴＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＦＯＲＴＨＥＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＰＲＯＢＬＥＭＯＦＡＨＩＧＨＯＲＤＥＲＥＬＬＩＰＴＩＣＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮ（刘国庆）（苏煜城）ＡＵＮＩＦＯ...     

A UNIFORMLY DIFFERENCE SCHEME OF SINGULAR PERTURBATION PROBLEM FOR A SEMILINEAR ORDINARY DIFFERENTIAL EQUATION WITH MIXED BOUNDARY VALUE CONDITION

ＡＵＮＩＦＯＲＭＬＹＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＯＦＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＰＲＯＢＬＥＭＦＯＲＡＳＥＭＩＬＩＮＥＡＲＯＲＤＩＮＡＲＹＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＷＩＴＨＭＩＸＥＤＢＯＵＮＤＡＲＹＶＡＬＵＥＣＯＮＤＩＴ...     

A uniformly convergent finite difference method for a singularly perturbed initial value problem

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INTERPOLATION PERTURBATION METHOD FOR SOLVING THE BOUNDARY LAYER TYPE PROBLEMS

ＹｕａｎＹｉｗｕ（袁镒吾）（ＲｅｃｅｉｖｅｄＯｃｔ．２，１９９４；ＣｏｍｍｕｎｉｃａｔｅｄｂｙＣｈｉｅｎＷｅｉｚａｎｇ）ＩＮＴＥＲＰＯＬＡＴＩＯＮＰＥＲＴＵＲＢＡＴＩＯＮＭＥＴＨＯＤＦＯＲＳＯＬＶＩＮＧＴＨＥＢＯＵＮＤＡＲＹＬＡＹＥＲＴＹＰＥＰＲＯＢ...     

Computer computation of the method of multiple scales—dirichlet problem for a class of system of nonlinear differential equations

The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations. The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales. Contributed by Jiang Fu-ru, Original Member of Editorial Committe, AMM Biography: Xie La-bing (1976∼); Jiang Fu-ru(1927∼)