Multi-symplectic wavelet splitting method for the strongly coupled Schrodinger system

<正>We propose a multi-symplectic wavelet splitting method to solve the strongly coupled nonlinear Schrodinger equations.Based on its multi-symplectic formulation,the strongly coupled nonlinear Schr(o|¨)dinger equations can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem.For the linear subsystem,the multi-symplectic wavelet collocation method and the symplectic Buler method are employed in spatial and temporal discretization,respectively.For the nonlinear subsystem,the mid-point symplectic scheme is used.Numerical simulations show the effectiveness of the proposed method during long-time numerical calculation.     

Second order conformal multi-symplectic method for the damped Korteweg–de Vries equation

A conformal multi-symplectic method has been proposed for the damped Korteweg–de Vries(DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissmann box scheme,we obtain a conformal multi-symplectic scheme for multi-symplectic partial differential equations(PDEs) with added dissipation. Applying it to the DKdV equation, we construct a conformal multi-symplectic algorithm for it, which is of second order accuracy in time. Numerical experiments demonstrate that the proposed method not only preserves the dissipation rate of mass exactly with periodic boundary conditions, but also has excellent long-time numerical behavior.     

Multi-symplectic scheme for the coupled Schrdinger-Boussinesq equations

In this paper, a multi-symplectic Hamiltonian formulation is presented for the coupled Schrdinger-Boussinesq equations (CSBE). Then, a multi-symplectic scheme of the CSBE is derived. The discrete conservation laws of the Langmuir plasmon number and total perturbed number density are also proved. Numerical experiments show that the multi-symplectic scheme simulates the solitary waves for a long time, and preserves the conservation laws well.     

A conservative Fourier pseudospectral algorithm for a coupled nonlinear Schroedinger system

We derive a new method for a coupled nonlinear Schr/Sdinger system by using the square of first-order Fourier spectral differentiation matrix D1 instead of traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove the proposed method preserves the charge and energy conservation laws exactly. In numerical tests, we display the accuracy of numerical solution and the role of the nonlinear coupling parameter in cases of soliton collisions. Numerical experiments also exhibit the excellent performance of the method in preserving the charge and energy conservation laws. These numerical results verify that the proposed method is both a charge-preserving and an energy-preserving algorithm.     

Multi-symplectic variational integrators for nonlinear Schrdinger equations with variable coefficients

In this paper, we propose a variational integrator for nonlinear Schrdinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrdinger equations with variable coefficients, cubic nonlinear Schrdinger equations and Gross–Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space.     

A conservative Fourier pseudospectral algorithm for a coupled nonlinear Schrdinger system

We derive a new method for a coupled nonlinear Schrdinger system by using the square of first-order Fourier spectral differentiation matrix D1 instead of traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative.We prove the proposed method preserves the charge and energy conservation laws exactly.In numerical tests,we display the accuracy of numerical solution and the role of the nonlinear coupling parameter in cases of soliton collisions.Numerical experiments also exhibit the excellent performance of the method in preserving the charge and energy conservation laws.These numerical results verify that the proposed method is both a charge-preserving and an energy-preserving algorithm.     

Simulation of shock-induced instability using an essentially conservative adaptive CE/SE method

An essentially conservative adaptive space time conservation element and solution element(CE/SE)method is proposed for the effective simulation of shock-induced instability with low computational cost.Its implementation is based on redefined conservation elements(CEs)and solution elements(SEs),optimized interpolations and a Courant number insensitive CE/SE scheme.This approach is used in two applications,the Woodward double Mach reflection and a twocomponent Richtmyer–Meshkov instability experiment.This scheme reveals the essential features of the investigated cases,captures small unstable structures,and yields a solution that is consistent with the results from experiments or other high order methods.     

A local energy-preserving scheme for Klein Gordon Schrdinger equations

A local energy conservation law is proposed for the Klein–Gordon–Schr ¨odinger equations, which is held in any local time–space region. The local property is independent of the boundary condition and more essential than the global energy conservation law. To develop a numerical method preserving the intrinsic properties as much as possible, we propose a local energy-preserving(LEP) scheme for the equations. The merit of the proposed scheme is that the local energy conservation law can hold exactly in any time–space region. With the periodic boundary conditions, the scheme also possesses the discrete change and global energy conservation laws. A nonlinear analysis shows that the LEP scheme converges to the exact solutions with order O(τ2+ h2). The theoretical properties are verified by numerical experiments.     

Novel energy dissipative method on the adaptive spatial discretization for the Allen–Cahn equation

We propose a novel energy dissipative method for the Allen–Cahn equation on nonuniform grids. For spatial discretization, the classical central difference method is utilized, while the average vector field method is applied for time discretization. Compared with the average vector field method on the uniform mesh, the proposed method can involve fewer grid points and achieve better numerical performance over long time simulation. This is due to the moving mesh method, which can concentrate the grid points more densely where the solution changes drastically. Numerical experiments are provided to illustrate the advantages of the proposed concrete adaptive energy dissipative scheme under large time and space steps over a long time.     

Local structure-preserving methods for the generalized Rosenau-RLW-KdV equation with power law nonlinearity

Local structure-preserving algorithms including multi-symplectic, local energy-and momentum-preserving schemes are proposed for the generalized Rosenau–RLW–Kd V equation based on the multi-symplectic Hamiltonian formula of the equation. Each of the present algorithms holds a discrete conservation law in any time–space region. For the original problem subjected to appropriate boundary conditions, these algorithms will be globally conservative. Discrete fast Fourier transform makes a significant improvement to the computational efficiency of schemes. Numerical results show that the proposed algorithms have satisfactory performance in providing an accurate solution and preserving the discrete invariants.     

Multi-Symplectic Wavelet Collocation Method for Maxwell's Equations

In this paper, we develop a multi-symplectic wavelet collocation method for three-dimensional (3-D) Maxwell's equations. For the multi-symplectic formulation of the equations, wavelet collocation method based on autocorrelation functions is applied for spatial discretization and appropriate symplectic scheme is employed for time integration. Theoretical analysis shows that the proposed method is multi-symplectic, unconditionally stable and energy-preserving under periodic boundary conditions. The numerical dispersion relation is investigated. Combined with splitting scheme, an explicit splitting symplectic wavelet collocation method is also constructed. Numerical experiments illustrate that the proposed methods are efficient, have high spatial accuracy and can preserve energy conservation laws exactly.     

A Multi-Symplectic Compact Method for the Two-Component Camassa-Holm Equation with Singular Solutions

The two-component Camassa-Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa-Holm equation. Based on its multi-symplectic formulation, the proposed method is derived by the sixth-order compact finite difference method in spatial discretization and the symplectic implicit midpoint scheme in temporal discretization. Numerical experiments finely describe the velocity and density variables in the two-component integrable system and distinctly display the evolvement of the singular solutions. Moreover, the proposed method shows good conservative properties during long-time numerical simulation.     

Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations

In this paper, we study the integration of Hamiltonian wave equations whose solutions have oscillatory behaviors in time and/or space. We are mainly concerned with the research for multi-symplectic extended Runge–Kutta–Nyström (ERKN) discretizations and the corresponding discrete conservation laws. We first show that the discretizations to the Hamiltonian wave equations using two symplectic ERKN methods in space and time respectively lead to an explicit multi-symplectic integrator (Eleap-frogI). Then we derive another multi-symplectic discretization using a symplectic ERKN method in time and a symplectic partitioned Runge–Kutta method, which is equivalent to the well-known Störmer–Verlet method in space (Eleap-frogII). These two new multi-symplectic schemes are extensions of the leap-frog method. The numerical stability and dispersive properties of the new schemes are analyzed. Numerical experiments with comparisons are presented, where the two new explicit multi-symplectic methods and the leap-frog method are applied to the linear wave equation and the Sine–Gordon equation. The numerical results confirm the superior performance and some significant advantages of our new integrators in the sense of structure preservation.     

The complex multi-symplectic scheme for the generalized sinh-Gordon equation

In this paper,the complex multi-symplectic method and the implementation of the generalized sinhGordon equation are investigated in detail.The multi-symplectic formulations of the generalized sinh-Gordon equation in Hamiltonian space are presented firstly.The complex method is introduced and a complex semi-implicit scheme with several discrete conservation laws(including a multi-symplectic conservation law(CLS),a local energy conservation law(ECL) as well as a local momentum conservation law(MCL)) is constr...     

Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation

In this paper, we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac (NLD) equation. Based on its multi-symplectic formulation, the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system. Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem, respectively. And the nonlinear subsystem is solved by a symplectic scheme. Finally, a composition method is applied to obtain the final schemes for the NLD equation. We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly. Numerical experiments are presented to show the effectiveness of the proposed methods.     

A multi-symplectic numerical integrator for the two-component Camassa–Holm equation

A new multi-symplectic formulation of the two-component Camassa-Holm equation (2CH) is presented, and the associated local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. A multi-symplectic discretisation based on this new formulation is exemplified by means of the Euler box scheme. Furthermore, this scheme preserves exactly two discrete versions of the Casimir functions of 2CH. Numerical experiments show that the proposed numerical scheme has good conservation properties.     

Multi-Symplectic Splitting Method for Two-Dimensional Nonlinear Schrödinger Equation

Using the idea of splitting numerical methods and the multi-symplectic methods, we propose a multi-symplectic splitting(MSS) method to solve the two-dimensional nonlinear Schrödinger equation (2D-NLSE) in this paper. It is further shown that the method constructed in this way preserve the global symplecticity exactly. Numerical experiments for the plane wave solution and singular solution of the 2D-NLSE show the accuracy and effectiveness
of the proposed method.     

Multi-Symplectic Method for the Zakharov-Kuznetsov Equation

A new scheme for the Zakharov-Kuznetsov (ZK) equation with the accuracy order of $\mathcal{O}(∆t^2+∆x+∆y^2)$ is proposed. The multi-symplectic conservation property of the new scheme is proved. The backward error analysis of the new multi-symplectic scheme is also implemented. The solitary wave evolution behaviors of the Zakharov-Kunetsov equation are investigated by the new multi-symplectic scheme. The accuracy of the scheme is analyzed.     

一维Gross-Pitaevskii方程的高阶紧致分裂步多辛格式

首先把一维Gross-Pitaevskli方程改写成多辛Hamiltonian系统的形式,把形式通过分裂变成2个子哈密尔顿系统.然后,对这些子系统用辛或者多辛算法进行离散.通过对子系统数值算法的不同组合方式,得到不同精度的具有多辛算法特征数值格式.这些格式不仅具有多辛格式、分裂步方法和高阶紧致格式的特征,而且是质量守恒的.数值实验验证了新格式的数值行为.