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.     

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.     

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．     

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.     

带五次项的非线性Schrdinger方程的多辛Fourier拟谱算法

本文构造了带五次项的非线性Schr(o|¨)dinger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.     

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.     

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．     

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.     

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.     

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.     

Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions

We consider for the integration of coupled nonlinear Schrödinger equations with periodic plane wave solutions a splitting method from the class of symplectic integrators and the multi-symplectic six-point scheme which is equivalent to the Preissman scheme. The numerical experiments show that both methods preserve very well the mass, energy and momentum in long-time evolution. The local errors in the energy are computed according to the discretizations in time and space for both methods. Due to its local nature, the multi-symplectic six-point scheme preserves the local invariants more accurately than the symplectic splitting method, but the global errors for conservation laws are almost the same.     

Decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations with smooth initial data

Based on the mixed finite element method, we consider the decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations with smooth initial data in this paper. The temporal treatment of the spatial discrete Boussinesq equations is based on the implicit Crank–Nicolson scheme for the linear terms and the explicit Adams–Bashforth scheme for the nonlinear terms. Thanks to the decoupled method, the considered problem is split into two subproblems and these subproblems can be solved in parallel. Under some restriction on the time step, we present the stability and convergence results of numerical solutions, Finally, some numerical experiments are provided to test the performance of the developed numerical scheme and verify the established theoretical findings.     

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.     

Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of tsunami waves

Considered here are Boussinesq systems of equations of surface water wave theory over a variable bottom. A simplified such Boussinesq system is derived and solved numerically by the standard Galerkin-finite element method. We study by numerical means the generation of tsunami waves due to bottom deformation and we compare the results with analytical solutions of the linearized Euler equations. Moreover, we study tsunami wave propagation in the case of the Java 2006 event, comparing the results of the Boussinesq model with those produced by the finite-difference code MOST, that solves the shallow water wave equations.     

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.     

Analysis and approximation of linear feedback control problems for the Boussinesq equations

The mathematical formulation and numerical solution of the linear feedback control problem associated with the Boussinesq equations are presented. We show that the unsteady solutions to the Boussinesq equations are stabilizable by internal controllers with exponential decaying property. Semidiscrete-in-time and full space-time discrete approximations are also studied. Some computational results are presented     

Convection in stars

This paper reviews work on stellar convection theory with a particular emphasis on numerical simulations of convection. The Boussinesq and the anelastic approximations of the fluid equations are introduced and solutions of the linearized equations for the onset of convection are described. Non-linear numerical solutions to the Boussinesq equations are discussed. Next, the character of stellar convection is considered. Convection is found to be intimately linked with a number of proceses in the stellar interior. Studies of stellar convection have either attempted to model a star with a very simplified treatment of convection or to solve the fluid equations for a range of parameters less extreme than those found in stars. Finally, a brief review is given of some of the numerical methods that have been employed.