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.     

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.     

非线性弦振动方程的多辛算法

利用Hamiltonian空间体系下的多辛理论研究了非线性弦微小横向振动问题的数值解法．基于Bridges意义下的多辛积分理论,首先推导了非线性弦振动方程的一阶多辛偏微分方程组及其多种守恒律,随后构造了一种等价于Box多辛格式的新隐式多辛格式,最后,运用该多辛格式对非线性弦振动方程进行了数值模拟,并将模拟结果与吕克璞等人得到的解析解进行比较．数值实验结果显示利用本文构造的多辛格式得到的数值解与吕克璞等人得到的解析解非常接近,这说明该多辛格式能够较为精确地模拟非线性弦振动问题,同时数值结果也反映出了多辛方法的两大优点：精确的保持多种守恒律和良好的长时间数值行为．     

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．     

Klein-Gordon方程的紧致辛中点格式

本文利用紧致算子和修正的辛中点格式构造了Klein-Gordon方程初值问题的保结构算法.该紧致辛中点格式在时间方向具有二阶精度,在空间方向具有六阶精度,保持离散的辛结构,是线性稳定的算法.另外,该算法保持线性系统的离散能量,而对非线性系统,该算法满足一个离散能量的转移公式.数值算例验证了理论分析.     

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.     

广义Zakharov-Kuznetsov方程的若干能量守恒算法

本文基于哈密尔顿偏微分方程的多辛形式,利用平均值离散梯度构造了若干二维广义Zakharov-Kuznetsov方程的能量守恒算法,包括一个局部能量守恒算法及一个整体能量守恒算法.并证明了在周期边界条件下,两个格式均保持离散整体能量.数值例子验证了方法的有效性及正确性.     

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.     

空间太阳能电站太阳能接收器二维展开过程的保结构分析

针对传统数值方法求解微分-代数方程过程中经常遇到的违约问题,本文以空间太阳能电站太阳能接收器的简化二维模型为例,采用辛算法模拟了简化模型的展开过程,研究了辛算法在求解过程中约束违约问题.首先,基于Hamilton变分原理,将描述简化二维模型展开过程的Euler-Lagrange方程导入Hamilton体系,建立其Hamilton正则方程;随后,采用s级PRK离散方法离散正则方程,得到其辛格式;最后,采用辛PRK格式模拟太阳能接收器的二维展开过程.模拟结果显示:本文构造的辛PRK格式能够很好地满足系统的位移约束.     

WBK方程稳态解的保结构分析

水动力学系统稳定状态下的总能量与系统初始能量之差直观反映了水力系统的水头损失.本文基于保结构思想,以色散浅水波WBK模型为例,推导了其对称形式及空间辛结构等守恒性质.随后,采用Euler Box差分离散方法构造对称形式的保结构差分格式,并推导其离散空间辛结构,为数值格式保结构性能检验提供理论依据.最后,通过数值实验,考察数值格式的保结构性能,并将数值格式用于研究不同相对扩散系数条件下,WBK方程保结构稳态水质点系统的总能量,为水力系统水头损失的分析提供参考.     

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.     

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.     

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.     

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 methods for the coupled 1D nonlinear Schrödinger system

In the paper, the multi-symplectic formulation of the coupled 1D nonlinear Schrödinger system (CNLS) is considered. For the multi-symplectic formulation, a new six point scheme, which is equivalent to the multi-symplectic Preissman integrator, is derived. We also present numerical experiments, which show that the multi-symplectic scheme has excellent long-time numerical behaviour and energy conservation property.     

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．     

Multi-symplectic integration of coupled non-linear Schrödinger system with soliton solutions

Systems of coupled non-linear Schrödinger equations with soliton solutions are integrated using the six-point scheme which is equivalent to the multi-symplectic Preissman scheme. The numerical dispersion relations are studied for the linearized equation. Numerical results for elastic and inelastic soliton collisions are presented. Numerical experiments confirm the excellent conservation of energy, momentum and norm in long-term computations and their relations to the qualitative behaviour of the soliton solutions.