膜受迫振动方程的多辛格式及其守恒律

基于Hamilton空间体系的多辛理论研究了膜强迫振动问题.利用Runge-Kutta多辛格式构造了一种9×3点半隐式的多辛离散格式,该格式满足多辛守恒律.数值算例结果表明该多辛离散格式不仅能够有效提高数值计算精度,而且能够保持膜振动系统的局部性质.同时利用多辛格式模拟得到的波形图表明多辛方法具有较好的长时间数值稳定性.     

一类二次KdV类型水波方程的多辛Fourier拟谱方法

二次KdV类型水波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.本文基于Hamilton系统的多辛理论研究了一类二次KdV类型水波方程的数值解法,利用Fourier拟谱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.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.     

变形Boussinesq方程组的多辛Preissmann格式计算研究

基于Hamilton空间体系的多辛理论,研究了变形Boussinesq方程组的数值解法. 利用Preissman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. 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.     

饱和非线性薛定谔方程多辛Euler-Box方法

对饱和非线性薛定谔方程构造了两个Euler—box格式并将它们组合成了一个新的多辛离散格式．利用新的多辛离散格式模拟饱和非线性薛定谔方程．数值结果表明新的多辛离散格式能够很好地模拟饱和非线性薛定谔方程中孤子波的演化行为,并能近似地保持系统的模平方守恒特性．     

谐振子的辛欧拉分析方法

针对理想简谐振子力学模型,研究了其守恒律,并利用辛欧拉格式分析简谐振子振动过程.首先给出了谐振子系统的平方守恒律、周期守恒律和相差守恒律.构造了谐振子的普通欧拉格式和辛欧拉格式,研究了两种格式下三种守恒律各自的保持情况.模拟结果显示:辛欧拉格式能够精确保持时域守恒律(平方守恒律),但无法保持频域守恒律(周期守恒律和相差守恒律).如要克服辛欧拉格式的不足,需按邢誉峰教授提出的方法进行校正.     

一种简化的三阶精度加权ENO格式

91．引言从七十年代后期开始，对双曲型守恒律方程数值方法的研究以VanLeer构造出来的MUSCL格式［‘］为先导，出现了一些全新的高分辨率守恒型差分格式．特别是A．Harten［‘］提出了TVD（TotalVariationDiminising）格式的概念后，双曲型守恒律方程数值方法的研究取得了飞速的发展．因为TVD格式可以保持数值解的单调性，所以它可以有效地抑制间断附近数值解的振荡，这方面有重要代表性的工作是［2－4］．由于TVD格式必须保证数值解的总变差不增，所以使得TVD格式在光滑解的局部极值点处降价．为了克服TVD格式的这个弱点，便出…     

大阻尼杆振动的广义多辛算法

本文基于Bridges教授建立的多辛算法理论及其Hamilton变分原理,采用广义多辛算法研究了大阻尼杆的阻尼振动特性.引入正交动量后,首先将描述大阻尼杆振动的控制方程降阶为一阶Hamilton近似对称形式,即广义多辛形式;随后采用中点离散方法构造形式广义多辛形式的中点Box广义多辛离散格式;最后通过计算机模拟研究大阻尼杆振动过程中的耗散效应.研究结果表明,本文构造的广义多辛算法不仅能够保持系统守恒型几何性质,同时能够再现系统的耗散效应.     

Qi系统的Hopf分叉分析与幅值控制

通过非线性状态反馈,不改变Hopf分叉点,实现对四维Qi系统极限环的幅值控制．推导出Qi系统在第一类非零平衡点上产生Hopf分叉的条件,绘制第一类平衡点的分叉图．采用washout filter非线性控制律,利用中心流形定理对受控系统降维,得到极限环的幅值与控制增益之间的近似解析式．通过数值模拟以及幅值解析解与数值解的比较,验证幅值预测的正确性与控制的有效性．     

求解强非线性动力系统响应的一种新方法

将同伦理论和参数变换技术相结合提出了一种可适用于求解强非线性动力系统响应的新方法．即PE-HAM方法(基于参数展开的同伦分析技术)．其主要思想是通过构造合适的同伦映射,将一非线性动力系统的求解问题,转化为一线性微分方程组的求解问题,然后借助于参数展开技术消除长期项,进而得到系统的解析近似解．为了检验所提方法的有效性,研究了具有精确周期的保守Duffing系统的响应,求出了其解析的近似解表达式．在与精确周期的比较中,可以得出：在非线性强度。很大,甚至在α→∞时,近似解的周期与原系统精确周期的误差也只有2．17％．数值模拟结果说明了新方法的有效性．     

含表面裂纹简支梁的非线性振动分析

对含表面裂纹简支梁在大幅振动下的几何非线性进行了理论分析,从建立了梁的非线性振动的半解析解.用Rayleigh方法将振型函数表示为线性模型振型函数的组合,建立了梁非线性振动的第一阶振型函数的显式表达式,数值模拟计算了不同的裂纹深度和给定不同第一函数系数a1对梁最大位移的影响.建立的显式方程简单,易于工程应用.     

Multi-symplectic method for peakon–antipeakon collision of quasi-Degasperis–Procesi equation

Focusing on the local geometric properties of the shockpeakon for the Degasperis–Procesi equation, a multi-symplectic method for the quasi-Degasperis–Procesi equation is proposed to reveal the jump discontinuity of the shockpeakon for the Degasperis–Procesi equation numerically in this paper. The main contribution of this paper lies in the following: (1) the uniform multi-symplectic structure of the b-family equation is constructed; (2) the stable jump discontinuity of the shockpeakon for the Degasperis–Procesi equation is reproduced by simulating the peakon–antipeakon collision process of the quasi-Degasperis–Procesi equation. First, the multi-symplectic structure and several local conservation laws are presented for the b-family equation with two exceptions (b=3$b=3$ and b=4$b=4$). And then, the Preissman Box multi-symplectic scheme for the multi-symplectic structure is constructed and the mathematical proofs for the discrete local conservation laws of the multi-symplectic structure are given. Finally, the numerical experiments on the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation are reported to investigate the jump discontinuity of shockpeakon of the Degasperis–Procesi equation. From the numerical results, it can be concluded that the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation can be simulated well by the multi-symplectic method and the simulation results can reveal the jump discontinuity of shockpeakon of the Degasperis–Procesi equation approximately.     

Multi-symplectic wavelet collocation method for the nonlinear Schrödinger equation and the Camassa–Holm equation

In this paper, we develop a novel multi-symplectic wavelet collocation method for solving multi-symplectic Hamiltonian system with periodic boundary conditions. Based on the autocorrelation function of Daubechies scaling functions, collocation method is conducted for the spatial discretization. The obtained semi-discrete system is proved to have semi-discrete multi-symplectic conservation laws and semi-discrete energy conservation laws. Then, appropriate symplectic scheme is applied for time integration, which leads to full-discrete multi-symplectic conservation laws. Numerical experiments for the nonlinear Schrödinger equation and Camassa–Holm equation show the high accuracy, effectiveness and good conservation properties of the proposed method.     

A semi-explicit multi-symplectic splitting scheme for a 3-coupled nonlinear Schrödinger equation

In this paper, we mainly propose an efficient semi-explicit multi-symplectic splitting scheme to solve a 3-coupled nonlinear Schrödinger (3-CNLS) equation. Based on its multi-symplectic formulation, the 3-CNLS equation can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, the multi-symplectic Fourier pseudospectral method and symplectic Euler method are employed in spatial and temporal discretizations, respectively. For the nonlinear subsystem, the mid-point symplectic scheme is used. Numerical experiments for the unstable plane waves show the effectiveness of the proposed method during long-time numerical calculation.     

A new multi-symplectic scheme for the generalized Kadomtsev–Petviashvili equation

We propose a new scheme for the generalized Kadomtsev–Petviashvili (KP) equation. The multi-symplectic conservation property of the new scheme is proved. Back error analysis shows that the new multi-symplectic scheme has second order accuracy in space and time. Numerical application on studying the KPI equation and the KPII equation are presented in detail.     

Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations

Conformal symplecticity is generalized to forced-damped multi-symplectic PDEs in 1 + 1 dimensions. Since a conformal multi-symplectic property has a concise form for these equations, numerical algorithms that preserve this property, from a modified equations point of view, are available. In effect, the modified equations for standard multi-symplectic methods and for space-time splitting methods satisfy a conformal multi-symplectic property, and the splitting schemes exactly preserve global symplecticity in a special case. It is also shown that the splitting schemes yield incorrect rates of energy/momentum dissipation, but this is not the case for standard multi-symplectic schemes. These methods work best for problems where the dissipation coefficients are small, and a forced-damped semi-linear wave equation is considered as an example.     

Multi-symplectic splitting method for the coupled nonlinear Schrödinger equation

In this paper, we propose a multi-symplectic splitting method to solve the coupled nonlinear Schrödinger (CNLS) equation by using the idea of splitting the multi-symplectic partial differential equation (PDE). Numerical experiments show that the proposed method can simulate the propagation and collision of solitons well. The corresponding errors in global energy and momentum are also presented to show the good preservation property of the proposed method during long-time numerical calculation.     

New schemes for the coupled nonlinear Schrödinger equation

In this paper, we present three new schemes for the coupled nonlinear Schrödinger equation. The three new schemes are multi-symplectic schemes that preserve the intrinsic geometry property of the equation. The three new schemes are also semi-explicit in the sense that they need not solve linear algebraic equations every time-step, which is usually the most expensive in numerical simulation of partial differential equations. Many numerical experiments on collisions of solitons are presented to show the efficiency of the new multi-symplectic schemes.     

带五次项的非线性Schrodinger方程的多辛Fourier拟谱算法

本文构造了带五次项的非线性Schodinger方程的多辛Fourier拟谱格式,并通过数值例子说明了该格式的有效性．