Maximum norm error bound of a linearized difference scheme for a coupled nonlinear Schrödinger equations

In this article, a linearized conservative difference scheme for a coupled nonlinear Schrödinger equations is studied. The discrete energy method and an useful technique are used to analyze the difference scheme. It is shown that the difference solution unconditionally converges to the exact solution with second order in the maximum norm. Numerical experiments are presented to support the theoretical results.     

A three‐level linearized finite difference scheme for the camassa–holm equation

The Camassa–Holm (CH) system is a strong nonlinear third‐order evolution equation. So far, the numerical methods for solving this problem are only a few. This article deals with the finite difference solution to the CH equation. A three‐level linearized finite difference scheme is derived. The scheme is proved to be conservative, uniquely solvable, and conditionally second‐order convergent in both time and space in the discrete L_∞ norm. Several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 451–471, 2014     

New conservative difference schemes for a coupled nonlinear Schrödinger system

In this paper, two conservative difference schemes for solving a coupled nonlinear Schrödinger (CNLS) system are numerically analyzed. Firstly, a nonlinear implicit two-level finite difference scheme for CNLS system is studied, then a linear three-level difference scheme for CNLS system is presented. An induction argument and the discrete energy method are used to prove the second-order convergence and unconditional stability of the linear scheme. Numerical examples show the efficiency of the new scheme.     

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     

High-order linear compact conservative method for the nonlinear Schrödinger equation coupled with the nonlinear Klein–Gordon equation

In this paper, we design a linear-compact conservative numerical scheme which preserves the original conservative properties to solve the Klein–Gordon–Schrödinger equation. The proposed scheme is based on using the finite difference method. The scheme is three-level and linear-implicit. Priori estimate and the convergence of the finite difference approximate solutions are discussed by the discrete energy method. Numerical results demonstrate that the present scheme is conservative, efficient and of high accuracy.     

Conservative numerical schemes for the Ostrovsky equation

The Ostrovsky equation describes gravity waves under the influence of Coriolis force. It is known that solutions of this equation conserve the L² norm and an energy function that is determined non-locally. In this paper we propose four conservative numerical schemes for this equation: a finite difference scheme and a pseudospectral scheme that conserve the norm, and the same types of schemes that conserve the energy. A numerical comparison of these schemes is also provided, which indicates that the energy conservative schemes perform better than the norm conservative schemes.     

一类带波动算子的非线性Schringer方程的一个守恒差分格式

该文对一类带波动算子的非线性 Schrodinger(NL S)方程提出了一个守恒的差分格式 ,证明了该格式的收敛性和稳定性 .数值计算结果表明 ,该格式对网比不敏感 ,具有很好的守恒性 ,并且比文 [1]中的不守恒格式提高了计算效率     

Finite difference discretization of the Benjamin‐Bona‐Mahony‐Burgers equation

Numerical solutions of the Benjamin‐Bona‐Mahony‐Burgers equation in one space dimension are considered using Crank‐Nicolson‐type finite difference method. Existence of solutions is shown by using the Brower's fixed point theorem. The stability and uniqueness of the corresponding methods are proved by the means of the discrete energy method. The convergence in L^∞‐norm of the difference solution is obtained. A conservative difference scheme is presented for the Benjamin‐Bona‐Mahony equation. Some numerical experiments have been conducted in order to validate the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Maximum error estimates for a compact difference scheme of the coupled nonlinear Schrödinger–Boussinesq equations

The coupled nonlinear Schrödinger–Boussinesq (SBq) equations describe the nonlinear development of modulational instabilities associated with Langmuir field amplitude coupled to intense electromagnetic wave in dispersive media such as plasmas. In this paper, we present a conservative compact difference scheme for the coupled SBq equations and analyze the conservative property and the existence of the scheme. Then we prove that the scheme is convergent with convergence order O(τ² + h⁴) in L_∞‐norm and is stable in L_∞‐norm. Numerical results verify the theoretical analysis.     

非线性Schringer方程的一个新的守恒差分格式

本文对非线性Ｓｃｈｒｏｄｉｎｇｅｒ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Ｚｈａｎｇ Ｆｅｉ等人（１９９５）的格式有较大幅度的提高．     

Conservative finite difference methods for fractional Schrödinger–Boussinesq equations and convergence analysis

In this paper, two conservative finite difference schemes for fractional Schrödinger–Boussinesq equations are formulated and investigated. The convergence of the nonlinear fully implicit scheme is established via discrete energy method, while the linear semi‐implicit scheme is analyzed by means of mathematical induction method. Our schemes are proved to preserve the total mass and energy in discrete level. The numerical results are given to confirm the theoretical analysis.     

广义非线性Schringer方程的一个新的守恒差分格式

对广义非线性Schro。d inger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C rank-N ico lson格式以及Zhang Fei等人提出的格式.     

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。     

非线性Schrdinger方程的高精度守恒差分格式

本文首先分析线性Schrodinger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Schrodinger方程,提出了一种精度为O(r2 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.     

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.     

On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation

In this paper, we study the initial-boundary value problem of the usual Rosenau-RLW equation by finite difference method. We design a conservative numerical scheme which preserves the original conservative properties for the equation. The scheme is three-level and linear-implicit. The unique solvability of numerical solutions has been shown. Priori estimate and second order convergence of the finite difference approximate solutions are discussed by discrete energy method. Numerical results demonstrate that the scheme is efficient and accurate.     

On Some Finite Difference Scheme for Gas Dynamics Equations

A conservative difference scheme with linear dependence of the pressure on the density of gas is proposed for gas dynamics equations. The scheme allows us to simulate 1-D flows inside a cylindrical domain with time-variable cross-sections and guarantees the positive sign of the density function.     

具有周期边界的守恒型方程的守恒型差分格式

研究守恒型奇摄动方程的周期边界问题,构造了一个守恒型差分格式,利用分解解的奇性项的方法,结合问题的渐近展开,证明所构造的差分格式为一阶一致收敛。     

Two-dimensional high-resolution schemes and their application in the modeling of ionizing waves in gas discharges

The method for constructing upwind high-resolution schemes is proposed in application to the modeling of ionizing waves in gas discharges. The flux-limiting criterion for continuity equations is derived using the proposed partial monotony property of a finite difference scheme. For two-dimensional extension, the cone transport upwind approach for constructing genuinely two-dimensional difference schemes is used. It is shown that when calculating rotations of symmetric profiles by using this scheme, a circular form of isolines is not distorted in a distinct from the coordinate splitting method. The conservative second order finite-difference scheme is proposed for solving the equations system of the drift-diffusion model of electric discharge; this scheme implies finite-difference conservation laws of electric charge and full electric current (fully conservative scheme). Computations demonstrate absence of numeric oscillations and good resolution of two-dimensional ionizing fronts in simulations of streamer and barrier discharges     

A completely conservative difference scheme of gas dynamics in orthogonal curvilinear coordinates

A completely conservative difference scheme in Eulerian curvilinear orthogonal coordinates is proposed for calculating discontinuous gas-dynamic flows on sufficiently coarse grids. The stability of the linear approximation of the proposed scheme is analyzed. Courant's condition provides a stability condition for this scheme.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 9–15, 1986.