一类偏积分微分方程二阶差分全离散格式

本给出了数值求解一类偏积分微分方程的二阶全离散差分格式．采用了Crank-Nicolson格式；积分项的离散利用了Lubieh的二阶卷积积分公式；给出了稳定性的证明，误差估计及收敛性的结果．     

On the convergence of finite difference schemes for the heat equation with concentrated capacity

Summary. We investigate the convergence of difference schemes for the one-dimensional heat equation when the coefficient at the time derivative (heat capacity) is represents the magnitude of the heat capacity concentrated at the point . An abstract operator method is developed for analyzing this equation. Estimates for the rate of convergence in special discrete energetic Sobolev's norms, compatible with the smoothness of the solution are obtained. Received November 2, 1999 / Revised version received July 24, 2000 / Published online May 4, 2001     

A second order difference scheme with nonuniform rectangular meshes for nonlinear parabolic system

In this paper, a difference scheme with nonuniform meshes is proposed for the initial-boundary problem of the nonlinear parabolic system. It is proved that the difference scheme is second order convergent in both space and time.     

A second order numerical scheme for the solution of the one-dimensional Boussinesq equation

A predictor–corrector (P-C) scheme is applied successfully to a nonlinear method arising from the use of rational approximants to the matrix-exponential term in a three-time level recurrence relation. The resulting nonlinear finite-difference scheme, which is analyzed for local truncation error and stability, is solved using a P-C scheme, in which the predictor and the corrector are explicit schemes of order 2. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behaviour of the P-C/MPC schemes is tested numerically on the Boussinesq equation already known from the bibliography free of boundary conditions. The numerical results are derived for both the bad and the good Boussinesq equation and conclusions from the relevant known results are derived.      

一类偏积分微分方程一阶差分全离散格式

本文给出了数值求解一类偏积分微分方程的一阶差分全离散格式。时间方向采用了一阶向后差分格式,空间方向采用二阶差分格式,给出了稳定性的证明,误差估计及收敛性的结果,并给出了数值例子。     

A difference scheme for Vlasov's equation

A difference scheme is developed for the solution of the initial-value problem for the one-dimensional Vlasov equation for a collisionless electron plasma with a homogeneous ion background. Instead of the distribution function itself, its Fourier transform with respect to velocity is used. Case studies of nonlinear Landau damping, a stationary equilibrium, and a two-stream instability are presented as preliminary applications.

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Zusammenfassung In dieser Arbeit wird ein Differenzenschema für die Lösung des Anfangswertproblems für die eindimensionale Vlasov-Gleichung für ein stoßfreies Elektronenplasma mit homogenem Ionenhintergrund entwickelt. Statt der Verteilungsfunktion selbst wird deren Fouriertransformierte bezüglich der Geschwindigkeitsvariablen benutzt. Als vorläufige Anwendungen werden Fallstudien über nichtlineare Landaudämpfung, ein stationäres Gleichgewicht und eine Zweistrominstabilität durchgeführt.

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This work was supported in part by the Atomic Energy Commission Grant No.AT(11-1)-2059.     

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 new explicit three-level difference scheme for the solution of the heat flow equation

A new explicit three-level difference scheme for the numerical solution of the heat flow equation is proposed. The main features of the new scheme are: i) it is unconditionally stable, ii) it is very highly accurate from the point of view of the truncation error, and iii) its solution converges to the solution of the heat equation even if the time and the distance increments tend to zero independently.     

High accuracy finite difference scheme for three-dimensional microscale heat equation

A compact finite difference scheme is developed to the three-dimensional microscale heat transport equation. This new scheme is fourth order in space and second order in time. It is proved to be unconditionally stable with respect to initial values. Numerical results are provided for comparison testing purpose.     

一类具有连续变量的二阶非线性阻尼差分方程的振动准则

考虑了一类具有连续变量的二阶非线性阻尼差分方程,利用积分变换和广义R iccati变换,给出了此类方程的振动准则.     

一类带弱奇异核偏积分微分方程二阶差分空间半离散格式的全局稳定性

本文给出了数值求解一类带弱奇异核偏积分微分方程的二阶差分空间半离散格式;借助于Laplace变换及Parseval等式,得到了全局稳定性的证明.     

五维热传导方程的一族两层显格式

构造了五维热传导方程的一族两层显格式,证明了当截断误差阶为O（τ＋h2）时,其稳定性条件为网比r=hτ2≤21,优于同类的其它显格式,当截断误差阶为O（τ2＋h2）时,可以得到一个简洁而实用的二阶精度的两层显格式.     

A difference scheme for the biharmonic equation with nonlinear boundary condition

In a rectangular domain we construct a grid scheme by applying the operators of exact difference schemes. We study an estimate of the rate of convergence of the grid scheme in the grid norm L₂(). It is shown that in the case when the solution of the differential problem belongs to the space W ₂ ^k (), k (3/2,2] the order of precision of the proposed scheme is O(h^k–3/2), and in the linear case it is O(h^k).Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 3–14.     

High accurate scheme for Boussinesq equation

In this paper, numerical methods for solving the good Boussinesq equation (G. B.) are considered. A fourth order accurate, difference scheme which satisfies the conservation laws is proposed. Under certain condition, the proof of stability, convergence and error estimation of this scheme is given. Some other problems related to Boussinesq are illustrated.     

Three positive solutions for a second order difference equation with three-point boundary value problem

In this paper, we consider the solutions of discrete second-order boundary value problem $$\left\{\begin{array}{l}\Delta ^{2}y(k-1)+a(k)f(k,y(k))=0,\quad k\in \{1,\ldots,T\},\\[2pt]y(0)-\alpha\Delta y(0)=0,\quad y(T+1)=\beta y(n),\end{array}\right.$$ where f is continuous, T≥3 is a fixed positive integer, n∈{2,…,T?1}, constant α,β>0 such that T+1?β n+α(1?β)>0,T+1?β n>0. Under suitable conditions, we accomplish this by using the property of the associate Green’s function and Leggett-Williams fixed point theorem.     

A variational difference scheme for the one-dimensional diffusion equation

In this paper we construct a homogeneous variational difference scheme for the diffusion equation assuming its coefficients to be bounded and measurable; the order of convergence of the scheme is O(h²). We consider the boundary value problem (1) $$\frac{d}{{dx}}\left( {K(x)\frac{{du}}{{dx}}} \right) - g(x)u = - \frac{{dF}}{{dx}},0< x< X$$ subject to the boundary conditions (2) $$u(0) = a,u(X) = b$$ .     

A compact difference scheme for the fractional diffusion-wave equation

This article is devoted to the study of high order difference methods for the fractional diffusion-wave equation. The time fractional derivatives are described in the Caputo’s sense. A compact difference scheme is presented and analyzed. It is shown that the difference scheme is unconditionally convergent and stable in _L∞$L_{\infty }$-norm. The convergence order is O(τ^3-α+h⁴)$O({\tau }^{3-\alpha }+h^4)$. Two numerical examples are also given to demonstrate the theoretical results.     

一类二阶差分方程泛函边值问题的多解性

考虑二阶差分方程泛函边值问题△2u(k-1)=(Fu)(k),k∈[a+1,b-1]z,ω(u)=A,γ(△u)=B多个解的存在性,并获得一个严格单调递增解和一个严格单调递减解.其中a,b∈Z,满足b≥a+2,F为连续算子,ω,γ均为连续泛函.