解对流方程的加耗散项的差分格式

解对流方程的大多数常见的显式差分格式 ,其稳定性条件是苛刻的 .这一困难可由在常规的显式差分格式中引入耗散项而得到克服 .基于此 ,我们导出一类新的无条件稳定的两层的半显式差分格式及若干具有高稳定性的显式格式 .它们包含了若干已知的具有高稳定性的显式格式 .     

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.     

耗散对称正则长波方程的有限差分逼近

本文对耗散对称正则长波方程的初边值问题进行了数值研究,提出了一个两层隐式Crank-Nicolson差分格式,讨论了差分解的存在唯一性,并利用能量方法分析了该格式的二阶收敛性与稳定性,数值算例表明本文的格式是可靠的.     

套网格有限差分方法求解——一阶双曲方程初边值问题的稳定性条件

本文讨论用套网格(Nesting grid)有限差分方法求解一阶双曲方程初边值问题,即在不同的区域做不同的网格剖分,选用相同或不同精度的差分格式.这种方法亦称杂交法(Hybrid Difference Method)或混合型差分方法(Mixed Difference Method),它广泛应用于数值天气预报和流体力学数值计算中,特别,对于局部区域上的解,其梯度变化激烈,而在其余区域上解的梯度变化平稳时,选用这种方法更有优越性。 [5,8,10]是套网格差分格式稳定性方面的工作,上述工作均以Kreiss定理为基础,针对两层显式耗散格式讨论,因而不便于应用,本文旨在利用GKS理论,寻求一般形式套网格差分格式稳定性的判别条件,§1针对模型问题建立套网格差分格式的一般形式,并介绍GKS理论的一种变形;§2建立套网格差分格式稳定性判别条件;§3是对一类差分格式和网格条件给出易于检验的稳定性判别准则;§4推广了Ciment匹配定理,并证明§3中的主要结果,最后§5是数值例子, 本文采用[1],[3]的符号。     

广义K.d.V.方程的数值方法

本文研究一般的广义K.d.V.方程的数值方法,给出了广义K.d.V.方程的一类半离散差分格式,证明了它们的守恒性。作者还严格证明了这类格式的广义稳定性,并由此推出收敛性。文章的最后考虑了全离散情形和两步格式。     

一类反应扩散方程组的保持正性的差分格式

本文研究一类具有正解的反应扩散方程组的有限差分解法.构造了一个保持正性的差分格式.利用离散的最大值原理证明了差分格式解的非负性,有界性及差分格式的无条件稳定性.这些估计的证明不依赖于微分方程的解而仅仅与初边值条件有关.当微分方程的解适当光滑时,证明了差分格式的一致收敛性.最后给出了数值计算结果,并与以往方法进行了比较.计算结果说明了本文给出的方法的有效性.     

求解非线性时滞脉冲双曲微分方程的隐式差分格式

构造了一类求解非线性时滞脉冲双曲型偏微分方程的隐式差分格式.在一定条件下,获得了该差分格式的唯一可解性、收敛性和无条件稳定性,且空间和时间均二阶精度.最后,数值实验表明了所得格式的精度和有效性.     

一类广义的Kdv方程的显式差分解

本文考察一类具耗散的广义KdV方程组■ (gradφ(■))_x ■_(xxx)-α■_(xx) γ■ =■(x,t,■)的周期初值问题的显式差分格式．利用有界延拓法证明了该差分格式的收敛性与稳定性,并给出了算法和数值例子．     

KdV-Burgers方程的一类本性并行差分方法

KdV-Burgers方程是非线性耗散和色散型波动方程,可以作为湍流规范方程,具有广泛的物理背景,其数值解法具有重要的科学意义和实际应用价值.针对KdV-Burgers方程,本文结合经典Crank-Nicolson格式和四个不同类型的Saul'yev非对称格式,提出了一类本性并行差分方法,构造交替分段Crank-Nicolson(ASC-N)差分格式.分析证明了ASC-N格式解的存在唯一性,线性绝对稳定性和计算精度.理论分析和数值试验结果均表明ASC-N差分格式线性绝对稳定,具有空间2阶精度,时间2阶精度(除内边界点外).在计算效率上,ASC-N格式具有明显的并行计算性质,相比较于隐式格式大幅度节省了计算时间.表明本文方法求解KdV-Burgers方程是高效可行的.     

带Gilbert耗散项Landau—Lifshitz方程组的有限差 分解

本文对一类带有Gilbert耗散项的Landau-Lifshitz铁磁链方程组第二边值问题建立了隐式有限差分格式.证明了差分解的存在性,并讨论了差分解的收敛性.     

On stability of numerical schemes via frozen coefficients and the magnetic induction equations

We study finite difference discretizations of initial boundary value problems for linear symmetric hyperbolic systems of equations in multiple space dimensions. The goal is to prove stability for SBP-SAT (Summation by Parts—Simultaneous Approximation Term) finite difference schemes for equations with variable coefficients. We show stability by providing a proof for the principle of frozen coefficients, i.e., showing that variable coefficient discretization is stable provided that all corresponding constant coefficient discretizations are stable.     

Stability of Implicit Difference Schemes with Space and Time-Dependent Coefficients

A stability theorem is derived for implicit difference schemes approximating multidimensional initial-value problems for linear hyperbolic systems with variable coefficients, and lots of widely used difference schemes are proved to be stable under the conditions similar to those for the cases of constant coefficients. This theorem is an extension of the stability theorem due to Lax-Nirenberg. The proof is quite simple.     

On the Asymptotical Stability of Multivariate Dynamical Systems

The systems considered are multivariate, linear, time–discrete, and time–variable. Parts of the asymptotical stability set in the parameter space spanned by the time–variable coefficients are explicitly found.     

Stability and convergence of difference schemes for boundary value problems for the fractional-order diffusion equation

A family of difference schemes for the fractional-order diffusion equation with variable coefficients is considered. By the method of energetic inequalities, a priori estimates are obtained for solutions of finite-difference problems, which imply the stability and convergence of the difference schemes considered. The validity of the results is confirmed by numerical calculations for test examples.     

一类半线性变系数抛物型方程的紧差分格式

对一类半线性变系数抛物型方程初边值问题建立了紧差分格式,用能量分析方法证明了差分格式解的存在唯一性、关于初值的无条件稳定性和在L_∞范数下阶数为O(τ~2+h~4)的收敛性,最后给出的数值算例验证了理论结果.     

A Nonlocal Finite-Difference Boundary-Value Problem

We investigate the stability of difference schemes for the equation of heat conduction with nonlocal boundary conditions. An example is given which in a certain sense imitates the problem with variable coefficients and has an exact solution in analytical form. It is shown that the difference operator has a simple spectrum and that multiple eigenvalues appear only in the case with constant coefficients. The simple spectrum ensures that the eigenvectors of the finite-difference problem form a basis. This enables us to apply to the nonlocal problem the theory of stability of symmetrizable difference schemes.     

Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients

One of the basic inverse problems in an anisotropic media is the determination of coefficients in a bounded domain with a single measurement. We consider the problem of finding the coefficient of the second derivatives in a second-order hyperbolic equation with variable coefficients.

Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement. Moreover we show that our uniqueness results yield the Lipschitz stability estimate in L ² space for solution to the inverse problem under consideration.     

Stability of nonautonomous difference equations with several delays

We consider the possibility to construct efficient stability criteria for solutions to difference equations with variable coefficients. We prove that one can associate a difference equation with a certain functional differential equation, whose solution has the same asymptotic behavior. We adduce examples, demonstrating the essential character of conditions of the obtained theorems and the exactness of the constant 3/2 which defines the boundary of the stability domain.     

Stability of semi-autonomous difference equations

For a linear difference equation with constant coefficients and several bounded variable delays we obtain criteria for the uniform and uniformly exponential stability expressed in terms of parameters of the initial problem. We adduce examples that prove the exactness of the boundaries of the obtained stability domain.     

Stability of an implicit difference scheme for a linear differential-algebraic system of partial differential equations

An implicit difference scheme is considered for approximating the initial-boundary value problem for a linear differential-algebraic system of partial differential equations with variable matrix coefficients of special structure. The conditional and asymptotic stability of the difference scheme with respect to the initial and boundary conditions and the right-hand side is proved.