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1.
This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete multi-symplectic conservation law to solve the partial differential equations which are derived from the generalized fifth-order KdV equation numerically. The results of the numerical experiments show that this multi-symplectic algorithm is good in accuracy and its long-time numerical behaviour is also perfect. 相似文献
2.
Yaming Chen Songhe Song & Huajun Zhu 《advances in applied mathematics and mechanics.》2014,6(4):494-514
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. 相似文献
3.
In this paper, we propose explicit multi-symplectic schemes for Klein–Gordon–Schrödinger equation by concatenating suitable symplectic Runge–Kutta-type methods and symplectic Runge–Kutta–Nyström-type methods for discretizing every partial derivative in each sub-equation. It is further shown that methods constructed in this way are multi-symplectic and preserve exactly the discrete charge conservation law provided appropriate boundary conditions. In the aim of the commonly practical applications, a novel 2-order one-parameter family of explicit multi-symplectic schemes through such concatenation is constructed, and the numerous numerical experiments and comparisons are presented to show the efficiency and some advantages of the our newly derived methods. Furthermore, some high-order explicit multi-symplectic schemes of such category are given as well, good performances and efficiencies and some significant advantages for preserving the important invariants are investigated by means of numerical experiments. 相似文献
4.
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. 相似文献
5.
We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler method for temporal discretization. It is verified that the proposed method has corresponding discrete multi-symplectic conservation laws. Numerical simulations indicate that the proposed scheme is characterized by excellent conservation. 相似文献
6.
Multi-symplectic variational integrators for nonlinear Schrdinger equations with variable coefficients
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In this paper, we propose a variational integrator for nonlinear Schrdinger 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 Schrdinger equations with variable coefficients, cubic nonlinear Schrdinger 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. 相似文献
7.
Yuto Miyatake Takaharu Yaguchi Takayasu Matsuo 《Journal of computational physics》2012,231(14):4542-4559
We consider structure preserving numerical schemes for the Ostrovsky equation, which describes gravity waves under the influence of Coriolis force. This equation has two associated invariants: an energy function and the L2 norm. It is widely accepted that structure preserving methods such as invariants-preserving and multi-symplectic integrators generally yield qualitatively better numerical results. In this paper we propose five geometric integrators for this equation: energy-preserving and norm-preserving finite difference and Galerkin schemes, and a multi-symplectic integrator based on a newly found multi-symplectic formulation. A numerical comparison of these schemes is provided, which indicates that the energy-preserving finite difference schemes are more advantageous than the other schemes. 相似文献
8.
A generalization of the multi-symplectic form for Hamiltonian systems to self-adjoint systems with dissipation terms is studied. These systems can be expressed as multi-symplectic Birkhoffian equations, which leads to a natural definition of Birkhoffian multi-symplectic structure. The concept of Birkhoffian multi-symplectic integrators for Birkhoffian PDEs is investigated. The Birkhoffian multi-symplectic structure is constructed by the continuous variational principle, and the Birkhoffian multi-symplectic integrator by the discrete variational principle. As an example, two Birkhoffian multi-symplectic integrators for the equation describing a linear damped string are given. 相似文献
9.
David Cohen Takayasu Matsuo Xavier Raynaud 《Journal of Nonlinear Mathematical Physics》2014,21(3):442-453
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. 相似文献
10.
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... 相似文献
11.
王俊杰 《原子与分子物理学报》2013,30(6)
广义Zakharov-Kuznetsov 方程作为一类重要的非线性方程有着许广泛的应
用前景,基于Hamilton 空间体系的多辛理论研究了广义Zakharov-Kuznetsov方程的数值
解法,讨论了利用Preissmann 方法构造离散多辛格式的途径, 并构造了一种典型的半隐
式的多辛格式, 该格式满足多辛守恒律、局部能量守恒律. 数值算例结果表明该多辛离
散格式具有较好的长时间数值稳定性. 相似文献
12.
In this paper,we propose a conformal momentum-preserving method to solve a damped nonlinear Schrodinger(DNLS) equation.Based on its damped multi-symplectic formulation,the DNLS system can be split into a Hamiltonian part and a dissipative part.For the Hamiltonian part,the average vector field(AVF) method and implicit midpoint method are employed in spatial and temporal discretizations,respectively.For the dissipative part,we can solve it exactly.The proposed method conserves the conformal momentum conservation law in any local time-space region.With periodic boundary conditions,this method also preserves the total conformal momentum and the dissipation rate of momentum exactly.Numerical experiments are presented to demonstrate the conservative properties of the proposed method. 相似文献
13.
14.
广义Zakharov-Kuznetsov方程作为一类重要的非线性方程有着许多广泛的应用前景,基于Hamilton空间体系的多辛理论研究了广义Zakharov-Kuznetsov方程的数值解法,讨论了利用Preissmann方法构造离散多辛格式的途径,并构造了一种典型的半隐式的多辛格式,该格式满足多辛守恒律、局部能量守恒律.数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性. 相似文献
15.
Haochen Li Jianqiang Sun & Mengzhao Qin 《advances in applied mathematics and mechanics.》2015,7(1):58-73
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. 相似文献
16.
Huajun Zhu Songhe Song & Yaming Chen 《advances in applied mathematics and mechanics.》2011,3(6):663-688
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. 相似文献
17.
We investigate the multi-symplectic Euler-box scheme for the nonlinear Schroedinger equation. Two new simple semi-explicit scheme are derived. A composition scheme based on the new derived schemes is also discussed. Some numerical results are reported to illustrate the efficiency of the new schemes. 相似文献
18.
A Multi-Symplectic Compact Method for the Two-Component Camassa-Holm Equation with Singular Solutions
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《中国物理快报》2017,(9)
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. 相似文献
19.
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. 相似文献
20.
In this work, the initial-boundary value problem of two-dimensional Cahn–Hilliard equation is considered. A class of fully discrete dissipative Fourier spectral schemes are proposed. Moreover, semi-implicit prediction–correction schemes are presented. The numerical simulations are performed to demonstrate the effectiveness of the proposed schemes. 相似文献