小参数在高阶导数项的椭圆型方程第一边值问题的一致收敛差分格式

本文讨论了曲边区域上小参数ε在高阶导数项的椭圆型方程第一边值问题,从一致收敛的必要条件出发构造了特殊的差分格式,证明了差分方程问题解的一致收敛性,估计了收敛的阶数,并讨论了差分方程解的渐近性态.     

四阶椭圆型方程奇异摄动问题的数值解

本文对一类四阶椭圆型方程奇异摄动问题建立了指数型拟合差分格式,并且证明了这种格式在能量范数意义下关于小参数ε的一致收敛性.最后,我们给出了数值结果.     

自共轭椭圆型偏微分方程奇异摄动问题的一致收敛差分解法

本文在矩形域内考虑高阶导数项含有小参数的自共轭椭圆型第一边值问题. 本文,我们应用渐近分析方法建立了一种新的差分格式,比较了差分方程的解与微分方程的解的渐近性态,并证明了解的一致收敛性.     

自共轭椭圆型方程的样条解法

本文从二元样条空间的理论出发,构造了一类新的差分格式,并利用它得到了一类自共轭椭圆型方程的样条解,并证明了这样的解的唯一性和收敛性问题.最后,给出了一个数值例子,说明了本方法是可行的.     

非线性抛物型初边值问题的差分-边界有限元耦合方法及其误差估计

一、引言边界元方法以其对于无界区域问题的独特有效性及其它一些性质,在工程技术和计算数学领域得到越来越广泛的重视、应用和研究.对于椭圆型边值问题,边界元方法的应用和理论研究已是硕果累累,对于发展型的初边值问题,近十年来,其理论研究在某些方面已取得了突破性进展,但仍有许多方面处于空白.发展型方程的边界元方法基本上分为三种类型:第一种类型是利用发展型方程的基本解导出发展型的边界积分方程;第二种类型是通过可逆积分变换将发展方程转化为椭圆型方程;第三种类型是对于时间变量采用差分离散化,将发展型方程转化成一组椭圆型方程.对于第一种类型方法的应用和理论研究已日臻完善.但对于第三种类型方法的理论分析尚属空白.本文研究第三种类型方法的应用及其误差分析,给出了数值计算格式和近似解的先验误差估计.     

具有边梁加固的板弯曲问题的变分-差分方法研究

具有边梁加固的板的弯曲问题,其平衡方程模型为四阶椭圆型偏微分方程的边值问题,其中的自然边界条件涉及到了沿板边的切线和法线方向的高阶导数,对于非均匀、变厚度的板,该问题还具有"变系数"的特点.由问题的变分模型入手,应用变分-差分方法构造了该边值问题的一个差分格式.由于该方法能够结合平衡方程模型中的边界条件以消除沿板边的高阶导数项,因而,所得差分算子仅仅依赖于板面网格结点,并且保持了差分算子的对称、正定性质.同时,将已得算法在计算机上进行了数值模拟,并与现有文献进行了对比计算.结果显示本文所给出的算法具有较高的精确度,该算法将可用于定量地揭示板与边梁之间相互作用的规律,为工程设计提供参考依据.     

一类高阶椭圆型方程的样条解法

从二元样条空间S2^1(△mn^(2))的理论出发，构造一类新的差分格式，并利用它得到了一类高阶椭圆型方程边值问题的样条解，并证明了这样的解的存在唯一性和收敛性问题．     

扩散系数为矩阵形式的两相混溶驱动问题的修正迎风差分格式

1　引　　言油藏数值模拟对油田开发意义重大 .两相不可压缩混溶驱动问题 ,其数学模型是一组非线性偏微分方程 ,其中的压力方程是一椭圆型方程 ,饱和度方程是一对流扩散方程 .由于对流为主的扩散方程具有双曲特性 ,中心差分格式虽关于空间步长具有二阶精度 ,但会产生数值弥散和非物理力学特性的数值振荡 ,使数值模拟失真 .特征方法与标准的有限差分方法结合起来可以较好地反映出对流扩散方程的一阶双曲特性 ,从而减少误差 ,提高计算精度[1 ] .在周期性假定下 ,美国数学家 Jim Douglas,Jr教授分别对压力方程采用混合元格式[2 ] 和五点差分…     

Pohozaev恒等式及其在非线性椭圆型方程中的应用

在非线性椭圆型偏微分方程的研究中,Pohozaev恒等式在研究非平凡解的存在性和非存在性时起着十分重要的作用.本文旨在介绍Pohozaev恒等式及其在非线性椭圆型问题研究中的应用.首先介绍有界区域和无界区域上几种典型的Pohozaev恒等式,并得到几类非线性椭圆型方程存在解的必要条件,进而得到对应的方程非平凡解的非存在性和存在性结果.其次将介绍非线性椭圆型方程的局部Pohozaev恒等式,由此证明非线性椭圆型微分方程近似解序列的紧性,并得到几类典型非线性椭圆型方程的无穷多解存在性.最后利用非线性椭圆型方程的局部Pohozaev恒等式来研究其波峰解,得到波峰解的局部唯一性,并由此判断波峰解的对称性等特征.     

椭圆型奇异摄动问题差分格式的一致收敛性分析

本文研究了一类椭圆型奇异摄动问题.利用Bakhvalov-Shishkin网格上的差分方法,获得了数值解一致一阶收敛于真解的结果.     

An alternating direction implicit scheme for discrete ordinate transport equations

This paper concerns a finite difference approximation of the discrete ordinate equations for the time-dependent linear transport equation posed in a multi-dimensional rectangular parallelepiped with partially reflecting walls. We present an unconditionally stable alternating direction implicit finite difference scheme, show how to solve the difference equations, and establish the following properties of the scheme.If a sequence of difference approximations is considered in which the time and space increments approach zero, then the corresponding sequence of solutions has a subsequence which converges continuously to a strong solution of the discrete ordinate equations. Provided that the time increment is sufficiently small, independently of the space and velocity increment sizes: the solution of the difference equations is bounded by an exponential function of time; in the subcritical case the coefficient of t in this exponential bound is zero or negative; and if the constituent functions are all nonnegative, then the solution of the difference equations will also be nonnegative. This last result implies a monotonicity principle for solutions of related difference problems.     

Numerical solutions for some coupled systems of nonlinear boundary value problems

Summary In the well-known Volterra-Lotka model concerning two competing species with diffusion, the densities of the species are governed by a coupled system of reaction diffusion equations. The aim of this paper is to present an iterative scheme for the steady state solutions of a finite difference system which corresponds to the coupled nonlinear boundary value problems. This iterative scheme is based on the method of upper-lower solutions which leads to two monotone sequences from some uncoupled linear systems. It is shown that each of the two sequences converges to a nontrivial solution of the discrete equations. The model under consideration may have one, two or three nonzero solutions and each of these solutions can be computed by a suitable choice of initial iteration. Numerical results are given for these solutions under both the Dirichlet boundary condition and the mixed type boundary condition.     

Difference spline scheme for modeling of convective flows using full Navier — Stokes equations

A difference scheme is constructed, in which enhanced stability is achieved by simultaneous solutions of the equations of motion, energy, and continuity. Spline approximations of spatial derivatives (with the original equations written in divergence form) substantially improve the accuracy of the scheme compared with the standard difference scheme using symmetric differences. The efficiency of the scheme is demonstrated for some problems of convective flow of compressible gas with lateral and bottom heating.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 38–45, 1993.     

A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations

We approximate the regular solutions of the incompressible Euler equations by the solution of ODEs on finite-dimensional spaces. Our approach combines Arnold’s interpretation of the solution of the Euler equations for incompressible and inviscid fluids as geodesics in the space of measure-preserving diffeomorphisms, and an extrinsic approximation of the equations of geodesics due to Brenier. Using recently developed semi-discrete optimal transport solvers, this approach yields a numerical scheme which is able to handle problems of realistic size in 2D. Our purpose in this article is to establish the convergence of this scheme towards regular solutions of the incompressible Euler equations, and to provide numerical experiments on a few simple test cases in 2D.     

Solutions of n-point boundary value problems associated with nonlinear summary difference equations

Variation of parameter methods play a fundamental rôle in understanding solutions of perturbed nonlinear differential as well as difference equations. This paper is devoted to the study of n-point boundary value problems associated with systems of nonlinear first-order summary difference equations by using the nonlinear variation of parameter methods. New variational formulae, which provide connections between the solutions of initial value problems and n-point boundary value problems, are obtained. An iterative scheme for computing approximated solutions of the boundary value problems is also provided.     

The Order of Convergence of Difference Schemes for Fractional Equations

In this paper, we continue our research on convergence of difference schemes for fractional differential equations. Using implicit difference scheme and explicit difference scheme, we have a deal with the full discretization of the solutions of fractional differential equations in time variables and get the order of convergence.     

Finite Difference Method with Nonuniform Meshes for Quasilinear Parabolic Systems

1.lnthestudyoftheprobleminphysics,mechanics,chemicalreactions,biologyandotherpracticalsciences,thelinearandnonlinearparabolicequationsandsystemsareappearedveryfrequently.Manynumericalinvestigationsinscientificandengineeringproblemsespeciallyinthelargescalecomputationalproblemsoftencontainthenumer-icalsolutionsofparabolicequationsandsystems.ThemethodwithunequalmeshstePSisnotavoidableinthesecomputations.Manyunexpectedandselfcontradictoryphe-nomenonraisingfromtheuseofunequalmeshstepscallourgreata…     

An implicit pseudospectral scheme to solve propagating fronts in reaction‐diffusion equations

Some Reaction‐Diffusion equations present solutions of the traveling wave form. In this work, we present an implicit numerical scheme based on finite difference originally proposed to solve hyperbolic equations. Then, this method is improved using a pseudospectral approach to discretize the spatial variable. The results prove that this new scheme is useful to solve equations of the parabolic type which presents traveling wave solutions. In particular, problems where a reduction in the number of discretization points and an increase of the size of the time step play an important role in their solution are considered. The implicit scheme presented involves the solution of linear systems only. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 86–105, 2016     

Coupled solutions and monotone iterative techniques for some nonlinear initial value problems on time scales

A unified approach to monotone iterative technique concerning the existence of coupled maximal and minimal solutions for dynamic IVP's on time scales is developed in the case of the nonlinear term involved admitting a splitting of a difference of two monotone functions. Besides pointing out the relevance of such problems in mathematical biology involving both differential and difference equations, the results involved prove to be useful in the sense of including several known results as well as some interesting new results in the monotone iterative scheme theory, especially in the case of the corresponding difference equations.     

一类变系数微分方程差分格式的收敛性(英文)

本文首先对一类变系数微分方程建立有限差分格式.然后利用矩阵的特征值和范数理论,讨论该格式解的收敛性和唯一性.通过数值算例,说明该格式既有效又便于模拟.并且文中所用方法还能用于高阶微分方程和某些非线性微分方程问题的研究.