一维非线性Schrdinger方程的两个无条件收敛的守恒紧致差分格式

本文对一维非线性Schrdinger方程给出两个紧致差分格式,运用能量方法和两个新的分析技巧证明格式关于离散质量和离散能量守恒,而且在最大模意义下无条件收敛.对非线性紧格式构造了一个新的迭代算法,证明了算法的收敛性,并在此基础上给出一个新的线性化紧格式.数值算例验证了理论分析的正确性,并通过外推进一步提高了数值解的精度.     

RLW-KdV方程的紧致有限差分格式

本文研究了RLW-KdV方程的一个三层线性紧致有限差分格式.该格式是质量守恒和能量守恒的,用离散能量法证明了差分格式的收敛性和稳定性.所建格式的收敛阶为O(τ~2+h~4).数值实验验证了该格式的有效性和可靠性.     

非线性Schrödinger方程的高精度守恒差分格式

本文首先分析线性Schr(o)dinger方程一种高阶差分格式的构造方法,得到方程的耗散项.在此基础上对三次非线性Schr(o)dinger方程,提出了一种精度为o(τ2+h2)的差分格式,证明了该格式保持了连续方程的两个守恒量,且是收敛的与稳定的.并通过数值例子与已有隐格式进行了比较,结果表明,本文格式在计算量类似的情况下,提高了数值精度.     

一类带波算子的非线性Schr?dinger方程的高精度守恒差分格式

对带波动算子的非线性Schr?dinger方程给出了一个新的高精度的守恒差分格式,证明了该格式满足守恒式,且是收敛稳定的.在数值实验中给出了数值计算的实验结果,通过计算表明这个格式的精度具有O(τ2+h4),且明显高于其他几种格式的精度.     

一维电磁波方程的四阶紧致差分格式

本文建立求解一维电磁波方程的四阶紧致差分格式,运用von Neumann法给出方法的稳定条件.运用能量法证明格式的收敛性.最后,数值例子验证了格式的有效性.     

一类非自共轭非线性Schrödinger方程的三层差分格式

本文对一类非自共轭非线性Schrodinger方程提出了一种三层差分格式,并证明了该格式的收敛性与稳定性.这种格式不需叠代,故计算速度比C-N格式快,数值计算结果表明,该格式是有效的和可靠的.     

非线性Schr(o)dinger方程的一个新的守恒差分格式

本文对非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程提出了一种新的带参数的守恒差分格式,并证明了该格式的收敛性与稳定性,通过数值计算获得如下结论,本文提出的差分格式在取适当的参数后,精度上比ＺｈａｎｇＦｅｉ等人（１９９５）的格式有较大幅度的提高。     

带五次项的非线性Schrödinger方程的守恒差分格式

本文研究带有五次项的非线性Schrödinger方程初边值问题的有限差分法,其中方程中二阶偏导数项的系数、五次项的系数及初值满足下面的条件（1.6）.针对此问题,我们研究了一个守恒差分格式,在条件（1.6）下,差分解的$L^{\infty}$模先验估计被得到.在此基础上,我们得到了差分解最优$L^2$模的误差估计.     

广义Rosenau-KdV方程的高精度守恒差分格式

对广义Rosenau-KdV方程提出一种在时间层和空间层上分别具有二阶和四阶精度的三层线性差分格式,所建格式是离散质量守恒和离散能量守恒的,利用离散能量法证明了差分格式的可解性、收敛性和稳定性.数值实验验证了该格式的精度和守恒性.     

非线性Schroedinger方程初边值问题的守恒数值格式

该文对非线性Ｓｃｈｒｏｅｄｉｎｇｅｒ方程提出了一种新的守恒差分格式,并证明了该格式的收敛性与稳定性,通过数值计算获得如下结论,提出的差分格式在取适当的参数后,精度上好于文（７）中的格式。     

Numerical study of a conservative weighted compact difference scheme for the symmetric regularized long wave equations

In this article, a new weighted and compact conservative difference scheme for the symmetric regularized long wave (SRLW) equations is considered. The new scheme is decoupled and linearized in practical computation, that is, at each time step only two tridiagonal systems of linear algebraic equations need to be solved. It is proved by the discrete energy method that the compact scheme is uniquely solvable, the convergence and stability of the difference scheme are obtained, and its numerical convergence order is in the ‐norm. Numerical experiment results show that the scheme is efficient and reliable.     

A three level linearized compact difference scheme for the Cahn-Hilliard equation

This article is devoted to the study of high order accuracy difference methods for the Cahn-Hilliard equation.A three level linearized compact difference scheme is derived.The unique solvability and unconditional convergence of the difference solution are proved.The convergence order is O(τ 2 + h 4 ) in the maximum norm.The mass conservation and the non-increase of the total energy are also verified.Some numerical examples are given to demonstrate the theoretical results.     

Optimal H2‐error estimates of conservative compact difference scheme for the Zakharov equation in two‐space dimension

In this paper, a conservative compact difference scheme is proposed for the two‐dimensional nonlinear Zakharov equation with periodic boundary condition and initial condition. The proposed scheme not only conserve the mass and energy in the discrete level but also are efficient in practical experiments because the Fast Fourier transform (FFT) can be used to speed up the numerical computation. By using the standard energy method and induction argument, we can establish rigorously the unconditional and optimal H²‐error estimates. Some numerical examples are provided to support our theoretical results and show the accuracy and efficiency of the new scheme.     

Conservative compact difference schemes for the coupled nonlinear schrödinger system

In this article, some conservative compact difference schemes are explored for the strongly coupled nonlinear schrödinger system. After transforming the scheme into matrix form, we prove the existence and uniqueness, convergence and stability of the difference solutions for one nonlinear scheme in the norm by using some techniques of matrix theory. Numerical results show that one nonlinear scheme is the most efficient of all the compact schemes constructed here. It allows much larger time steps than the others. The second most efficient compact scheme is a linear one. We then give numerical simulations to two soliton interactions for the two most efficient compact schemes. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 749–772, 2014     

A conservative compact difference scheme for the coupled Klein–Gordon–Schrödinger equation

In this article, a conservative compact difference scheme is presented for the periodic initial‐value problem of Klein–Gordon–Schrödinger equation. On the basis of some inequalities about norms and the priori estimates, convergence of the difference solution is proved with order O(h⁴ +τ ²) in maximum norm. Numerical experiments demonstrate the accuracy and efficiency of the compact scheme. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Fourth‐order compact difference schemes for 1D nonlinear Kuramoto–Tsuzuki equation

In this article, first, we establish some compact finite difference schemes of fourth‐order for 1D nonlinear Kuramoto–Tsuzuki equation with Neumann boundary conditions in two boundary points. Then, we provide numerical analysis for one nonlinear compact scheme by transforming the nonlinear compact scheme into matrix form. And using some novel techniques on the specific matrix emerged in this kind of boundary conditions, we obtain the priori estimates and prove the convergence in norm. Next, we analyze the convergence and stability for one of the linearized compact schemes. To obtain the maximum estimate of the numerical solutions of the linearized compact scheme, we use the mathematical induction method. The treatment is that the convergence in norm is obtained as well as the maximum estimate, further the convergence in norm. Finally, numerical experiments demonstrate the theoretical results and show that one of the linearized compact schemes is more accurate, efficient and robust than the others and the previous. It is worthwhile that the compact difference methods presented here can be extended to 2D case. As an example, we present one nonlinear compact scheme for 2D Ginzburg–Landau equation and numerical tests show that the method is accurate and effective. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2080–2109, 2015     

The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model

The numerical simulation of the dynamics of the molecular beam epitaxy (MBE) growth is considered in this article. The governing equation is a nonlinear evolutionary equation that is of linear fourth order derivative term and nonlinear second order derivative term in space. The main purpose of this work is to construct and analyze two linearized finite difference schemes for solving the MBE model. The linearized backward Euler difference scheme and the linearized Crank‐Nicolson difference scheme are derived. The unique solvability, unconditional stability and convergence are proved. The linearized Euler scheme is convergent with the convergence order of O(τ + h²) and linearized Crank‐Nicolson scheme is convergent with the convergence order of O(τ² + h²) in discrete L²‐norm, respectively. Numerical stability with respect to the initial conditions is also obtained for both schemes. Numerical experiments are carried out to demonstrate the theoretical analysis. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Implicit compact difference schemes for the fractional cable equation

In this paper, we propose two implicit compact difference schemes for the fractional cable equation. The first scheme is proved to be stable and convergent in l_∞-norm with the convergence order O(τ + h⁴) by the energy method, where new inner products defined in this paper gives great convenience for the theoretical analysis. Numerical experiments are presented to demonstrate the accuracy and effectiveness of the two compact schemes. The computational results show that the two new schemes proposed in this paper are more accurate and effective than the previous.     

广义非线性Schroedinger方程的一个新的守恒差分格式

对广义非线性Schroedinger方程提出了一种新的差分格式。揭示了该差分格式满足两个守恒律，并证明该格式的收敛性和稳定性．数值实验结果表明，新的差分格式优于Crank-Nicolson格式以及Zhang Fei等人提出的格式。     

A finite difference scheme for the nonlinear time-fractional partial integro-differential equation

In this paper, a finite difference scheme is proposed for solving the nonlinear time-fractional integro-differential equation. This model involves two nonlocal terms in time, ie, a Caputo time-fractional derivative and an integral term with memory. The existence of numerical solutions is shown by the Leray-Schauder theorem. And we obtain the discrete L₂ stability and convergence with second order in time and space by the discrete energy method. Then the uniqueness of numerical solutions is derived. Moreover, an iterative algorithm is designed for solving the derived nonlinear system. Numerical examples are presented to validate the theoretical findings and the efficiency of the proposed algorithm.