Burgers方程的一类交替分组方法

对于Burgers方程给出了一组新的Saul'yev型非对称差分格式，并用这些差分格式构造了求解非线性Burgers方程的交替分组四点方法．该算法把剖分节点分成若干组，在每组上构造能够独立求解的差分方程．因此算法具有并行本性，能直接在并行计算机上使用．章还证明了所给算法线性绝对稳定．数值试验表明，该方法使用简便，稳定性好，有很好的精度。     

时间分数阶反应-扩散方程混合差分格式的并行计算方法

分数阶反应-扩散方程有深刻的物理和工程背景,其数值方法的研究具有重要的科学意义和应用价值.文中提出时间分数阶反应-扩散方程混合差分格式的并行计算方法,构造了一类交替分段显-隐格式(alternative segment explicit-implicit,ASE-I)和交替分段隐-显格式(alternative segment implicit-explicit,ASI-E),这类并行差分格式是基于Saul'yev非对称格式与古典显式差分格式和古典隐式差分格式的有效组合.理论分析格式解的存在唯一性,无条件稳定性和收敛性.数值试验验证了理论分析,表明ASE-I格式和ASI-E格式具有理想的计算精度和明显的并行计算性质,证实了这类并行差分方法求解时间分数阶反应-扩散方程是有效的.     

KdV-Burgers方程的一类新本性并行差分格式北大核心CSCD

KdV-Burgers方程作为湍流规范方程,具有深刻的物理背景,其快速数值解法具有重要的实际应用价值.针对KdV-Burgers方程,提出了一种新型的并行差分格式.基于交替分段技术,结合经典Crank-Nicolson(C-N)格式、显格式和隐格式,构造了混合交替分段Crank-Nicolson(MASC-N)差分格式.理论分析表明MASC-N格式是唯一可解、线性绝对稳定和二阶收敛的.数值试验表明,MASC-N格式比C-N格式具有更高的精度和效率.与ASE-I和ASC-N差分格式相比,MASC-N并行差分格式有最好的性能.表明该文的MASC-N并行差分方法能有效地求解KdV-Burgers方程.     

钝体绕流非定常解

本文结合目前流场显示的研究课题,对方块物体和山形物体的钝体绕流在起动阶段的运动情况,进行相应的数值模拟.并用有限差分方法求解二维不可压缩流体运动的N-S方程的非定常解.对差分格式中的显式,隐式和交替方向隐式几种格式进行了讨论.最后用显式和交替方向隐式方法计算了山形物体和方块物体在起动阶段的运动情况.     

Burgers-Fisher方程改进的交替分段Crank-Nicolson并行差分方法

Burgers-Fisher方程在气体动力学,热传导,弹性力学等领域有着广泛的应用,其快速数值解法具有重要的科学意义和工程应用价值.文中提出Burgers-Fisher方程改进的交替分段Crank-Nicolson(IASC-N)并行差分方法.IASC-N格式的构造是基于交替分段技术,将古典显式格式,隐式格式和Cran...     

对流扩散方程的数值计算

本文研究了对流扩散方程的一种并行格式.利用一组saul'yev型非对称格式进行二次构造,分别得到了一类并行GE格式和GEL、GER格式;进一步推广,得到绝对稳定的交替分组显式AGE格式,并用数值例子检验AGE格式的数值计算效果.     

对流扩散方程的高效稳定差分格式

基于二阶修正Dennis格式 ,提出了采用时间相关法求解定常对流扩散方程的一种具有节省内存空间和提高定常解收敛速度的有理式型优化半隐和松驰半隐紧致格式 .本文建立的差分格式具有运算量小、无网格雷诺数限制的优点 ,是无条件稳定和无条件单调的。通过对非线性Burgers方程进行的数值计算结果表明 ,文中构造的有理式型优化半隐和松驰半隐紧致格式适合于非线性问题计算 ,且保持了无条件稳定和无条件单调的特性 ,尤其能使定常解收敛速度加快 ,精度提高 .     

有限差分-配点法求解二维Burgers方程

将重心插值配点法结合Crank-Nicolson差分格式来求解Burgers方程.首先,利用Hopf-Cole变换将Burgers方程转化为线性热传导方程;空间方向采用重心插值配点法进行离散,时间方向采用Crank-Nicolson格式离散,导出对应的线性代数方程组,并对此计算格式进行相容性分析;最后,通过数值算例验证此计算格式具有高精度和有效性.     

色散方程的一类新的并行交替分段隐格式

本文给出了一组逼近色散方程的非对称差分格式，并用这组格式和对称的Crank-Nicolson型格式构造了求解色散方程的并行交替分段差分隐格式．这个格式是无条件稳定的，能直接在并行计算机上使用．数值试验表明，这个格式有很好的精度．     

基于卷积神经网络模型数值求解双曲型偏微分方程的研究

人工神经网络近年来得到了快速发展,将此方法应用于数值求解偏微分方程是学者们关注的热点问题.相比于传统方法其具有应用范围广泛（即同一种模型可用于求解多种类型方程）、网格剖分条件要求低等优势,并且能够利用训练好的模型直接计算区域中任意点的数值.该文基于卷积神经网络模型,对传统有限体积法格式中的权重系数进行优化,以得到在粗粒度网格下具有较高精度的新数值格式,从而更适用于复杂问题的求解.该网络模型可以准确、有效地求解Burgers方程和level set方程,数值结果稳定,且具有较高数值精度.     

Long-Time Behavior of Finite Difference Solutions of a Nonlinear Schrödinger Equation with Weakly Damped

A weakly damped Schrödinger equation possessing a global attractor are considered. The dynamical properties of a class of finite difference scheme are analysed. The existence of global attractor is proved for the discrete system. The stability of the difference scheme and the error estimate of the difference solution are obtained in the autonomous system case. Finally, long-time stability and convergence of the class of finite difference scheme also are analysed in the nonautonomous system case.     

LONG-TIME BEHAVIOR OF FINITE DIFFERENCE SOLUTIONS OF A NONLINEAR SCHRODINGER EQUATION WITH WEAKLY DAMPED

1. IntroductionThe nonlinear schr~r equation with weakly dampedwhere t = N, o > 0, together with appropriate boUndary and hatal condition, is ared inmany physical fields. The echtence of an attractor is one of the most boortant ~eristiCSfor a dissipative system. The long-tabs dynamics is completely determined by the attractorof the system. J.M. Ghidaglia[1] studied the lOng-the behavior of the nonlineaz Sequation (1.1) and proved the eAstence of a compact global attractor A in H'(n) which…     

提高反应—扩散方程有限差分格式的稳定性问题

This paper deals with the special nonlinear reaction-diffusion equation.The finite difference scheme with incremental unknowns approximating to the differential equation (2.1) is set up by means of introducing incremental unknowns methods.Through the stability analyzing for the scheme,it was shown that the stability conditions of the finite difference schemes with the incremental unknowns are greatly improved when compared with the stability conditions of the corresponding classic difference scheme.     

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.     

Numerical solutions of diffusion mathematical models with delay

In this paper an explicit numerical difference scheme for mixed problems for the delay diffusion equation is proposed, as a generalization of the classic difference scheme for the diffusion problem. A sufficient condition for the asymptotic stability of the new scheme is proved. Consistence, convergence and some properties of stability for this scheme are studied. Illustrative examples of numerical results are also included.     

二维半线性反应扩散方程的交替方向隐格式

本文研究一类二维半线性反应扩散方程的差分方法.构造了一个二层线性化交替方向隐格式.利用离散能量估计方法证明了差分格式解的存在唯一性、差分格式在离散H~1模下的二阶收敛性和稳定性.最后给出两个数值例子验证了理论分析结果.     

A Finite Difference Method for the Model of Wheezes

1.IntroductionInordertostudythepitchofwheezesinpatients,J.B.Grotbergandothershavegivenaclassofmathematicalmodelof.he....l1'2]:WherebandVaretheLaplaceoperatorandgradientoperator,respectively.TheCartesiancomponents(u,w)arethedimen-sionlessaxialfluidvelocityanddimensionlessverticalfluidvelocityrespectively.4(x,z)t)isthevelocitypotentialfunction,Pisthedi-mensionlessfluidDressuredeterminedfromtheunsteadyBernoul1iequation(1.3),Paisthesteadydrivingpressure,I.istheexternalpressure.M,Ai,B,gandTar…     

A three-time-level explicit difference scheme of the second order of accuracy for parabolic equations

The difference schemes of Richardson [1] and of Crank-Nicolson [2] are schemes providing second-order approximation. Richardson's three-time-level difference scheme is explicit but unstable and the Crank-Nicolson two-time-level difference scheme is stable but implicit. Explicit numerical methods are preferable for parallel computations. In this paper, an explicit three-time-level difference scheme of the second order of accuracy is constructed for parabolic equations by combining Richardson's scheme with that of Crank-Nicolson. Restrictions on the time step required for the stability of the proposed difference scheme are similar to those that are necessary for the stability of the two-time-level explicit difference scheme, but the former are slightly less onerous.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 751–759, November, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00489 and by the International Science Foundation under grants No. N8Q300 and No. JBR100.     

A Note on the Stability of the Integral-Differential Equation of the Hyperbolic Type in a Hilbert Space

In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The stability estimates for the solution of this problem are obtained. The difference scheme approximately solving this problem is presented. The stability estimates for the solution of this difference scheme are obtained. In applications, the stability estimates for the solutions of the nonlocal boundary problem for one-dimensional integral-differential equation of the hyperbolic type with two dependent limits and of the local boundary problem for multidimensional integral-differential equation of the hyperbolic type with two dependent limits are obtained. The difference schemes for solving these two problems are presented. The stability estimates for the solutions of these difference schemes are obtained.     

On the stable difference scheme for source identification nonlocal elliptic problem

Approximation of source identification problem for elliptic equation with integral-type nonlocal condition is discussed. The first order of accuracy difference scheme for elliptic nonlocal identification problem is studied. By using spectral resolution of a self-adjoint operator, we establish stability inequalities for solution of constructed scheme. Subsequently, the difference scheme for approximate solution of multidimensional boundary value problem with integral-type nonlocal and first kind boundary conditions is investigated on stability. Numerical test examples are presented.