Multi-symplectic method for generalized fifth-order KdV equation

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.

New explicit multi-symplectic scheme for nonlinear wave equation

Based on splitting multi-symplectic structures, a new multi-symplectic scheme is proposed and applied to a nonlinear wave equation. The explicit multi-symplectic scheme of the nonlinear wave equation is obtained, and the corresponding multi-symplectic conservation property is proved. The backward error analysis shows that the explicit multi-symplectic scheme has good accuracy. The sine-Gordon equation and the Klein-Gordon equation are simulated by an explicit multi-symplectic scheme. The numerical results show that the new explicit multi-symplectic scheme can well simulate the solitary wave behaviors of the nonlinear wave equation and approximately preserve the relative energy error of the equation.

Explicit multi-symplectic method for the Zakharov-Kuznetsov equation

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.

Multi-symplectic methods for membrane free vibration equation

In this paper, the multi-symplectic formulations of the membrane free vibration equation with periodic boundary conditions in Hamilton space are considered. The complex method is introduced and a semi-implicit twenty-seven-points scheme with certain discrete conservation laws-a multi-symplectic conservation law （CLS）, a local energy conservation law （ECL） as well as a local momentum conservation law （MCL） --is constructed to discrete the PDEs that are derived from the membrane free vibration equation. The results of the numerical experiments show that the multi-symplectic scheme has excellent long-time numerical behavior,

Multi-symplectic Runge-Kutta methods for Landau-Ginzburg-Higgs equation

Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations （PDEs） derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.

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

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

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 variational integrators for nonlinear Schrdinger equations with variable coefficients

In this paper, we propose a variational integrator for nonlinear Schrodinger 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 Schrodinger equations with variable coefficients, cubic nonlinear Schrodinger 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.

Multi-symplectic scheme for the coupled Schrdinger-Boussinesq equations

In this paper, a multi-symplectic Hamiltonian formulation is presented for the coupled Schrdinger-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.

Multi-symplectic method for the coupled Schrdinger–KdV equations

In this paper, we present a multi-symplectic Hamiltonian formulation of the coupled Schrtidinger-KdV equations （CS＇KE） with periodic boundary conditions. Then we develop a novel multi-symplectic Fourier pseudospectral （MSFP） scheme for the CSKE. In numerical experiments, we compare the MSFP method with the Crank-Nicholson （CN） method. Our results show high accuracy, effectiveness, and good ability of conserving the invariants of the MSFP method.

Multi-symplectic method for generalized Boussinesq equation

The generalized Boussinesq equation that represents a group of important nonlinear equations possesses many interesting properties. Multi-symplectic formulations of the generalized Boussinesq equation in the Hamilton space are introduced in this paper. And then an implicit multi-symplectic scheme equivalent to the multi-symplectic Box scheme is constructed to solve the partial differential equations （PDEs） derived from the generalized Boussinesq equation. Finally, the numerical experiments on the soliton solutions of the generalized Boussinesq equation are reported. The results show that the multi-symplectic method is an efficient algorithm with excellent long-time numerical behaviors for nonlinear partial differential equations.

THE MULTI-SYMPLECTIC ALGORITHM FOR "GOOD" BOUSSINESQ EQUATION

The multi-symplectic formulations of the 'Good' Boussinesq equation were considered. For the multi-symplectic formulation, a new fifteen-point difference scheme which is equivalent to the multi-symplectic Preissman integrator was derived. The numerical experiments show that, the multi- symplectic scheme have excellent long-time numerical. behavior.

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

We propose a multi-symplectic wavelet splitting equations. Based on its mu]ti-symplectic formulation, method to solve the strongly coupled nonlinear SchrSdinger the strongly coupled nonlinear SchrSdinger 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 Euler 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.

Explicit Multi-symplectic Method for a High Order Wave Equation of KdV Type

In this paper, we consider multi-symplectic Fourier pseudospectral method for a high order integrable equation of KdV type, which describes many important physical phenomena. The multi-symplectic structure are constructed for the equation, and the conservation laws of the continuous equation are presented. The multisymplectic discretization of each formulation is exemplified by the multi-symplectic Fourier pseudospectral scheme. The numerical experiments are given, and the results verify the efficiency of the Fourier pseudospectral method.

Multi-symplectic Geometry and Preissmann Scheme for GSDBM Equation

The multi-symplectic geometry for the GSDBM equation is presented in this paper. The multi-symplectic formulations for the GSDBM equation are presented and the local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. The multi-symplectic discretization of each formulation is exemplified by the multisymplectic Preissmann scheme. The numerical experiments are given, and the results verify the efficiency of the Preissmann scheme.

A Review on Stochastic Multi-symplectic Methods for Stochastic Maxwell Equations

Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by Hong et al.(A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268:255-268, 2014). Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.

Numerical Dispersion Relation of Multi-symplectic Runge-Kutta Methods for Hamiltonian PDEs

Numerical dispersion relation of the multi-symplectic Runge-Kutta （MSRK） method for linear Hamiltonian PDEs is derived in the present paper, which is shown to be a discrete counterpart to that possessed by the differential equation. This provides further understanding of MSRK methods. However, much still remains to be investigated further.

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.

Multi-symplectic method for the generalized(2+1)-dimensionalKdV-mKdV equation

In the present paper, a general solution involv- ing three arbitrary functions for the generalized （2＋1）- dimensional KdV-mKdV equation, which is derived from the generalized （1＋1）-dimensional KdV-mKdV equa- tion, is first introduced by means of the Wiess, Tabor, Carnevale （WTC） truncation method. And then multi- symplectic formulations with several conservation laws taken into account are presented for the generalized （2＋1）- dimensional KdV-mKdV equation based on the multi- symplectic theory of Bridges. Subsequently, in order to simulate the periodic wave solutions in terms of rational functions of the Jacobi elliptic functions derived from thegeneral solution, a semi-implicit multi-symplectic scheme is constructed that is equivalent 1：o the Preissmann scheme. From the results of the numerical experiments, we can con- clude that the multi-symplectic schemes can accurately sim- ulate the periodic wave solutions of the generalized （2＋1）- dimensional KdV-mKdV equation while preserve approxi- mately the conservation laws.

LOCAL STRUCTURE-PRESERVING ALGORITHMS FOR THE KLEIN-GORDON-ZAKHAROV EQUATION

In this paper, using the concatenating method, a series of local structure-preserving algorithms are obtained for the Klein-Gordon-Zakharov equation, including four multisymplectic algorithms, four local energy-preserving algorithms, four local momentumpreserving algorithms;of these, local energy-preserving and momentum-preserving algorithms have not been studied before. The local structure-preserving algorithms mentioned above are more widely used than the global structure-preserving algorithms, since local preservation algorithms can be preserved in any time and space domains, which overcomes the defect that global preservation algorithms are limited to boundary conditions. In particular, under appropriate boundary conditions, local preservation laws are global preservation laws.Numerical experiments conducted can support the theoretical analysis well.