Multisymplectic Lie group variational integrator for a geometrically exact beam in R3

In this paper we develop, study, and test a Lie group multisymplectic integrator for geometrically exact beams based on the covariant Lagrangian formulation. We exploit the multisymplectic character of the integrator to analyze the energy and momentum map conservations associated to the temporal and spatial discrete evolutions.     

Multisymplectic Fourier pseudo-spectral integrators for Klein-Gordon-Schrödinger equations

A multisymplectic Fourier pseudo-spectral scheme,which exactly preserves the discrete multisymplectic conservation law,is presented to solve the Klein-Gordon-Schrdinger equations.The scheme is of spectral accuracy in space and of second order in time.The scheme preserves the discrete multisymplectic conservation law and the charge conservation law.Moreover,the residuals of some other conservation laws are derived for the geometric numerical integrator.Extensive numerical simulations illustrate the numerical behavior of the multisymplectic scheme,and demonstrate the correctness of the theoretical analysis.     

Multisymplecticity of the centred box scheme for a class of hamiltonian PDEs and an application to quasi-periodically solitary waves

In this paper, the multisymplectic integrator for a class of Hamiltonian PDEs depending explicitly on time and spatial variables (nonautonomous Hamiltonian PDEs) is defined, and the multisymplecticity of the centred box scheme for this kind of Hamiltonian PDEs is proven. We give an application of the result to (periodic) quasi-periodic variable coefficient Korteweg-de Vries (qpKdV) equation, which is known to have a physical application in the propagation of surface waves in straits or channels with quasi-periodic varying depth and width in the time direction. We derive a multisymplectic scheme for a qpKdV equation in terms of the multisymplecticity of the centred box scheme, then make use of it to simulate numerically the (periodically) quasi-periodically solitary wave of the equation. Numerical experiments are presented in illustration of the multisymplectic scheme of qpKdV equation stemming the centred box discretization.     

Multisymplectic schemes for strongly coupled schrödinger system

In this paper, two semi-explicit multisymplectic schemes are derived for the strongly coupled schrödinger system. Based on the two new multisymplectic schemes, we obtain a multisymplectic composition scheme which improves the accuracy in time. The best merits of the present schemes are all implemented easily. Some numerical simulations are done for investigating nonlinear coupling and linear coupling. Numerical results indicate that the new multisymplectic composition scheme is effective.     

Multisymplectic Structure and Multisymplectic Scheme for the Nonlinear Ware Equation

Abstract The multisvmplectic structure of the nonlinear wave equation is derived directly from the variationalprinciple. In the numerical aspect,we present a multisymplectic nine points scheme which is equivalent to themultisymplectic Preissman scheme.A series of numerical results are reported to illustrate the effectiveness ofthe scheme.     

MULTISYMPLECTIC FOURIER PSEUDOSPECTRAL METHOD FOR THE NONLINEAR SCHRODINGER EQUATIONS WITH WAVE OPERATOR

In this paper, the multisymplectic Fourier pseudospectral scheme for initial-boundary value problems of nonlinear SchrSdinger equations with wave operator is considered. We investigate the local and global conservation properties of the multisymplectic discretization based on Fourier pseudospectral approximations. The local and global spatial conservation of energy is proved. The error estimates of local energy conservation law are also derived. Numerical experiments are presented to verify the theoretical predications.     

Multisymplectic variational integrators for PDEs of geometrically exact beam dynamics using algorithmic differentiation

For the simulation of geometrically exact beam dynamics [4], a multisymplectic Lie-group variational integrator [3] is derived. Based on the implementation of the discrete Lagrangian, algorithmic differentiation is used in the computation of both, the discrete Euler-Lagrange equations, and the Jacobi matrix needed for the Newton-Raphson iteration. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A multisymplectic explicit scheme for the modified regularized long-wave equation

In this paper, we derive a new 10-point multisymplectic scheme for the modified regularized long-wave equation. The new scheme is an explicit scheme in the sense that the third time level does not include nonlinear terms. Numerical results indicate that the new scheme not only provides satisfied numerical solutions, but also preserves three invariants of motion very well.     

Concatenating construction of the multisymplectic schemes for 2+1-dimensional sine-Gordon equation

In this paper, taking the 2+1-dimensional sine-Gordon equation as an example, we present the concatenating method to construct the multisymplectic schemes. The-method is to discretizee independently the PDEs in different directions with symplectic schemes, so that the multisymplectic schemes can be constructed by concatenating those symplectic schemes. By this method, we can construct multisymplectic schemes, including some widely used schemes with an accuracy of any order. The numerical simulation on the collisions of solitons are also proposed to illustrate the efficiency of the multisymplectic schemes.     

A Dissipation-Preserving Integrator for Damped Oscillatory Hamiltonian Systems

In this paper, based on discrete gradient, a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is established. The solution of this system is a damped nonlinear oscillator. Basically, lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this approach. The new integrator gives a discrete analogue of the dissipation property of the original system. Meanwhile, since the integrator is based on the variation-of-constants formula for oscillatory systems, it preserves the oscillatory structure of the system. Some properties of the new integrator are derived. The convergence is analyzed for the implicit iterations based on the discrete gradient integrator, and it turns out that the convergence of the implicit iterations based on the new integrator is independent of $\|M\|$, where $M$ governs the main oscillation of the system and usually $\|M\|\gg1$. This significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian system. Numerical experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature.     

Error analysis of discrete conservation laws for Hamiltonian PDEs under the central box discretizations

The central box scheme has been the most successful of the multisymplectic integrators for Hamiltonian PDEs. In this paper, we investigate conservative properties of the central box scheme for Hamiltonian PDEs and derive the error formulas of discrete local and global conservation laws of energy and momentum. We apply these results to the nonlinear Schrödinger equation and Klein-Gordon equation. Numerical experiments are presented to verify the theoretical predications.     

Error estimates of structure-preserving Fourier pseudospectral methods for the fractional Schrödinger equation

This paper gives a rigorous error analysis of the multisymplectic Fourier pseudospectral method for the nonlinear fractional Schrödinger equation. The method preserves some intrinsic structure properties including the generalized multisymplectic conservation law. By rewriting it in a matrix form similar to that in the finite difference method, the method is shown to be convergent in the discrete L² norm with the second-order accuracy in time and spectral accuracy in space. The key techniques in the analysis include the discrete energy method, cutoff of the nonlinearity, and a posterior bound of numerical solutions by using the inverse inequality. In a similar line, the convergence result for the symplectic Fourier pseudospectral method can also be established. Moreover, the errors in the local and global energy conservation laws of discrete systems are also investigated. Numerical tests are performed to confirm the theoretical results.     

关于一类高阶非线性系统的几乎干扰解耦问题

本文讨论了一类 SISO高阶下三角形非线性系统的几乎干扰解耦的反馈设计问题 ,对一般的由 L2 m- L2 m p(m =1,2 ,… )所定义的非线性增益指标 ,利用加幂积分器的新技巧和 Backstepping方法 ,构造性的给出了设计光滑静态反馈控制律的方法 ,在保证闭环系统具有内稳定的基础上 ,使系统达到干扰衰减 ,本文的结果不仅推广了加幂积分器技巧的应用 ,而且改进了相关文献的结果 .     

Lobatto IIIA-IIIB discretization of the strongly coupled nonlinear Schrödinger equation

In this paper, we construct a second order semi-explicit multi-symplectic integrator for the strongly coupled nonlinear Schrödinger equation based on the two-stage Lobatto IIIA-IIIB partitioned Runge-Kutta method. Numerical results for different solitary wave solutions including elastic and inelastic collisions, fusion of two solitons and with periodic solutions confirm the excellent long time behavior of the multi-symplectic integrator by preserving global energy, momentum and mass.     

An exact penalty method for solving optimal control problems

This paper discusses an algorithm for solving optimal control problems. An optimal control problem is presented where the final time is unknown. The algorithm consists of an integrator and a minimizer; the latter is an exact penalty function used to solve constrained nonlinear programming problems. Essentially, the optimal control problem is converted to a mathematical programming problem such that a point satisfying the differential equations via the integrator is provided to the minimizer, a lower performance index is obtained, the integrator is reinitiated, etc., until a suitable stopping criterion is satisfied.     

An exponential wave integrator Fourier pseudospectral method for the nonlinear Schrödinger equation with wave operator

In this article, an exponential wave integrator Fourier pseudospectral (EWI-FP) method is proposed for solving the nonlinear Schrödinger equation with wave operator. The numerical method is based on a Deuflhard-type exponential wave integrator for temporal integration and the Fourier pseudospectral method for spatial discretizations. The scheme is fully explicit and very efficient thanks to the fast Fourier transform. Numerical analysis of the proposed EWI-FP method is carried out and rigorous error estimates are established by means of the mathematical induction. Numerical results are reported to confirm the theoretical studies.     

MULTISYMPLECTIC COMPOSITION INTEGRATORS OF HIGH ORDER

A composition method for constructing high order multisymplectic integrators is presented in this paper. The basic idea is to apply composition method to both the time and the space directions. We also obtain a general formula for composition method.     

Energy preserving integration of bi-Hamiltonian partial differential equations

The energy preserving average vector field (AVF) integrator is applied to evolutionary partial differential equations (PDEs) in bi-Hamiltonian form with nonconstant Poisson structures. Numerical results for the Korteweg de Vries (KdV) equation and for the Ito type coupled KdV equation confirm the long term preservation of the Hamiltonians and Casimir integrals, which is essential in simulating waves and solitons. Dispersive properties of the AVF integrator are investigated for the linearized equations to examine the nonlinear dynamics after discretization.     

不确定非线性系统的鲁棒适应H_∞几乎干扰解耦问题

本文研究了带有不确定参数的非线性系统的鲁棒适应 H∞ 控制的几乎干扰解耦问题 .运用改进的加幂积分器技巧与递归设计方法 ,构造性地设计了一种光滑鲁棒动态反馈控制律 ,在保证闭环系统内稳定的基础上 ,使系统达到干扰衰减 .