对称正则长波方程的一个守恒差分格式

本文考虑了具有齐次边界条件的对称正则长波方程的有限差分法．构造了一个两层守恒的有限差分格式,利用离散泛函分析方法分析了格式的收敛性和稳定性,从理论上得到了收敛阶为O(h~2 τ)．数值试验表明,我们的方法是可信的．     

广义对称正则长波方程的守恒差分格式

文章考虑了具有齐次边界条件的广义对称正则长波方程的有限差分格式.提出了一个守恒并且线性非耦合的三层有限差分格式,由于格式在计算中只需要解三对角线性方程组,从而避免了其中的迭代计算.文中先讨论了一个离散守恒量,然后我们利用离散泛函分析方法证明了格式的收敛性和稳定性,从理论上得到了收敛阶为O(h~2+τ~2).通过数值试验表明,所提的方法是可靠有效的.     

Long Time Asymptotic Behavior of Solution of Difference Scheme for a Semilinear Parabolic Equation

In this paper we prove that the solution of implicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the corresponding nonlinear stationary problem as $t\rightarrow\infty$. For the discrete solution of nonlinear parabolic problem, we get its long time asymptotic behavior which is similar to that of the continuous solution. For simplicity, we consider one-dimensional problem.     

非线性波动方程的显式差分法

利用有界延拓法,研究了非线性波动方程周期初边值问题的显式差分解的收敛性与稳定性,避免了较难的先验估计,并放宽了非线性项的条件。     

求解广义正则长波方程的守恒差分格式

本文对广义正则长波方程的初边值问题提出了—个隐式差分格式，该格式合理地模拟了方程本身所具有的两个守恒律．给出了差分解的先验估计，证明了差分解的唯一可解性、无条件收敛性及其稳定性．     

Analysis of an Implicit Finite Difference Scheme for Time Fractional Diffusion Equation

Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α∈(0, 1). In this paper, an implicit finite difference scheme for solving the time fractional diffusion equation with source term is presented and analyzed, where the fractional derivative is described in the Caputo sense. Stability and convergence of this scheme are rigorously established by a Fourier analysis. And using numerical experiments illustrates the accuracy and effectiveness of the scheme mentioned in this paper.     

非线性Sobolev-Galpern方程有限差分格式在t＞0时的长时间收敛性和稳定性

本文讨论了非线性Sobolev-Galpern初边值问题,给出了Sobolev-Galpern方程的有限差分格式在t>0时的长时间收敛性和稳定性的证明．     

Stability and Convergence of an Implicit Difference Approximation for the Space Riesz Fractional Reaction-Dispersion Equation

In this paper,we consider a Riesz space-fractional reaction-dispersion equation (RSFRDE).The RSFRDE is obtained from the classical reaction-dispersion equation by replacing the second-order space derivative with a Riesz derivative of orderβ∈(1,2]. We propose an implicit finite difference approximation for RSFRDE.The stability and convergence of the finite difference approximations are analyzed.Numerical results are found in good agreement with the theoretical analysis.     

Convergence of a Linearized and Conservative Difference Scheme for the Klein-Gordon-Zakharov Equation

A linearized and conservative finite difference scheme is presented for the initial-boundary value problem of the Klein-Gordon-Zakharov (KGZ) equation. The new scheme is also decoupled in computation, whichmeans that no iteration is needed and parallel computation can be used, so it is expected to be more efficient in implementation. The existence of the difference solution is proved by Browder fixed point theorem. Besides the standard energy method, in order to overcome the difficulty in obtaining a priori estimate, an induction argument is used to prove that the new scheme is uniquely solvable and second order convergent for U in the discrete L^∞- norm, and for N in the discrete L^2-norm, respectively, where U and N are the numerical solutions of the KGZ equation. Numerical results verify the theoretical analysis.     

Fourth Order Symmetric Finite Difference Schemes for the Acoustic Wave Equation

We present an explicit, symmetric finite difference scheme for the acoustic wave equation on a rectangle with Neumann and/or Dirichlet boundary conditions. The scheme is fourth order accurate both in time and space. It is obtained by mass lumping of a finite element scheme. The accuracy and the difference approximations at the boundary are analyzed in terms of local and global errors. AMS subject classification (2000) 65M10     

A New Compact Finite Difference Method for Solving the Generalized Long Wave Equation

The aim of this article is to analyze a new compact finite difference method (CFDM) for solving the generalized regularized long wave (GRLW) equation. This method leads to a system of linear equations involving tridiagonal matrices and the rate of convergence of the method is of order O(k ² + h ⁴) where k and h are mesh sizes of time and space variables, respectively. Stability analysis of the method is investigated by the energy method and an error estimate is given. The propagation of single solitons and interaction of two solitary waves are applied to validate the method which is found to be accurate and efficient. Three invariants of the motion are evaluated to determine conservation properties of the method.     

Finite difference schemes for the two‐dimensional semilinear wave equation

In this article, two finite difference schemes for solving the semilinear wave equation are proposed. The unique solvability and the stability are discussed. The second‐order accuracy convergence in both time and space in the discrete H¹‐norm for the two proposed difference schemes is proved. Numerical experiments are performed to support our theoretical results.     

广义Rosenau-Burgers方程的差分算法

对广义Rosenau-Burgers方程的初边值问题进行了数值研究,提出了新的两层隐式差分格式,得到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无条件稳定性,并且给出数值算例进行验证.     

一类非线性反应-扩散方程有限差分格式的稳定性研究

In the article,the fully discrete finite difference scheme for a type of nonlinear reaction-diffusion equation is established.Then the new function space is introduced and the stability problem for the finite difference scheme is discussed by means of variational approximation method in this function space.The approach used is of a simple characteristic in gaining the stability condition of the scheme.     

半线性抛物型方程奇异摄动问题的数值解

本文讨论具有抛物边界层的半线性抛物型方程奇异摄动问题的数值解法,在非均匀网格上构造了两层非线性差分格式,证明了差分格式是一致收敛的,给出了一些数值例子.     

一类半线性波动方程的Sobolev指数

Gaustavo Ponce与Thomas C.Sideris猜测：对一些具有特殊非线性项的半线性波动方程,如utt-△u=u^k(Du)^αx∈R^n,k∈Z^ ,ρ＝│α│≥2,其中Sobloev指数会在[n/2,n/2 1]中,他们在x∈R^3时回答了这一问题,本文在R^n(n≥4）中得到了半线性波动方程utt-△u=u^k(Du)^α（x∈R^n,k∈R^n,k∈Z^ ,p=│α│≥2）的Sobolev指数为max{n/2,(n/2-1)1-3/l-1 2},此数确实在区间[n/2,n/2 1]中,特别当ρ≤n-1时,我们得到了此半线性波动方程的Sobolev指数为n/2。     

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.     

Scattering for Semilinear Wave Equation with Small Data in High Space Dimensions

In this paper we study the scattering theory for the semilincar wave equation u_{tt} - Δu = F(u(t, x), Du(t, x)) in R^n (n ≥ 4) with smooth and small data. We show that the scattering operator exists for the nonlinear term F = F(λ) = O(|λ|^{1, α}), where α is an integer and satisfies α ≥ 2, n = 4; α ≥ I, n ≥ 5.