New Conservative Schemes for Regularized Long Wave Equation

In this paper, two finite difference schemes are presented for initial-boundary value problems of Regularized Long-Wave(RLW) equation. They all have the advantages that there are discrete energies which are conserved. Convergence and stability of difference solutions with order O(h2+τ2) are proved in the energy norm. Numerical experiment results demonstrate the effectiveness of the proposed schemes.

Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

In this paper,two fourth-order compact finite difference schemes are derived to solve the nonlinear fourth-order wave equation which can be viewed as a generalized model from the nonlinear beam equation.Differing from the existing compact finite difference schemes which preserve the total energy in a recursive sense,the new schemes are proved to per-fectly preserve the total energy in the discrete sense.By using the standard energy method and the cut-off function technique,the optimal error estimates of the numerical solutions are established,and the convergence rates are of O(h^(4)+τ^(2))with mesh-size h and time-step τ.In order to improve the computational efficiency,an iterative algorithm is proposed as the outer solver and the double sweep method for pentadiagonal linear algebraic equations is introduced as the inner solver to solve the nonlinear difference schemes at each time step.The convergence of the iterative algorithm is also rigorously analyzed.Several numerical results are carried out to test the error estimates and conservative properties.

A New Framework of Convergence Analysis for Solving the General Nonlinear Schrodinger Equation using the Fourier Pseudo-Spectral Method in Two Dimensions

This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.

Error Estimate of a New Conservative Finite Difference Scheme for the Klein-Gordon-Dirac System

In this paper,we derive and analyze a conservative Crank-Nicolson-type finite difference scheme for the Klein-Gordon-Dirac(KGD)system.Differing from the derivation of the existing numerical methods given in literature where the numerical schemes are proposed by directly discretizing the KGD system,we translate the KGD equations into an equivalent system by introducing an auxiliary function,then derive a nonlinear Crank-Nicolson-type finite difference scheme for solving the equivalent system.The scheme perfectly inherits the mass and energy conservative properties possessed by the KGD,while the energy preserved by the existing conservative numerical schemes expressed by two-level’s solution at each time step.By using energy method together with the‘cut-off’function technique,we establish the optimal error estimate of the numerical solution,and the convergence rate is O(τ^(2)+h^(2))in l∞-norm with time stepτand mesh size h.Numerical experiments are carried out to support our theoretical conclusions.

Unconditional convergence of a linearized implicit finite difference method for the 2D/3D Gross-Pitaevskii equation with angular momentum rotation

This paper is concerned with the time-step condition of a linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a 'lifting' technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schr¨odinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.

EFFICIENT AND ACCURATE NUMERICAL METHODS FOR LONG-WAVE SHORT-WAVE INTERACTION EQUATIONS IN THE SEMICLASSICAL LIMIT REGIME

This paper focuses on performance of several efficient and accurate numerical methods for the long-wave short-wave interaction equations in the semiclassical limit regime. The key features of the proposed methods are based on:(i) the utilization of the first-order or second-order time-splitting method to the nonlinear wave interaction equations;(ii) the ap-plication of Fourier pseudo-spectral method or compact finite difference approximation to the linear subproblem and the spatial derivatives;(iii) the adoption of the exact integration of the nonlinear subproblems and the ordinary differential equations in the phase space. The numerical methods under study are efficient, unconditionally stable and higher-order accurate, they are proved to preserve two invariants including the position density in L^1. Numerical results are reported for case studies with different types of initial data, these results verify the conservation laws in the discrete sense, show the dependence of the numerical solution on the time-step, mesh-size and dispersion parameter ε, and demonstrate the behavior of nonlinear dispersive waves in the semi-classical limit regime.

Existence of Generalized Heteroclinic Solutions of the Coupled Schrdinger System under a Small Perturbation

The following coupled Schrdinger system with a small perturbation uxx + u- u3+ βuv2+ f(, u, ux, v, vx) = 0 in R,vxx- v + v3+ βu2v + g(, u, ux, v, vx) = 0 in R is considered, where β and are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and its dominant system has a heteroclinic solution. Then adjusting some appropriate constants and applying the fixed point theorem and the perturbation method yield that this heteroclinic solution deforms to a heteroclinic solution exponentially approaching the obtained periodic solution(called the generalized heteroclinic solution thereafter).