厄尔尼诺和南方涛动海气耦合模型中参数估计的变分方法

提出一种基于变分原理估计厄尔尼诺和南方涛动海气耦合模型中未知参数的方法. 首先将所研究的非线性海气耦合动力方程引入到目标泛函中; 接着利用变分方法导出伴随方程和待辨识参数泛函梯度的公式; 然后设计了估计未知参数的算法.数值试验结果表明变分方法是一 种能有效估计海气耦合非线性系统未知参数的方法.     

修正的变分迭代法在四阶Cahn-Hilliard方程和BBM-Burgers方程中的应用

变分迭代法是一种基于变分原理,具有高数值精度的数值格式,目前已广泛应用于各类强非线性孤立波方程的数值求解中.本文利用修正的变分迭代法对两类非线性方程进行研究.该格式是对原数值方法的一种改进,即在变分项前引入了参数h.通过定义误差函数的离散二范数并在定义域内绘出h-曲线,从而确定出使误差达到最小的h,再返回原迭代过程进行求解.同时,参数的引入也扩大了原数值解的收敛域,在迭代次数一定的情况下达到了数值最优.在数值实验中,将上述结果应用于四阶的Cahn-Hilliard方程和BenjaminBona-Mahoney-Burgers方程.对于四阶的Cahn-Hilliard方程,普通的变分迭代法绝对误差在10~(-1)左右,经过修正后,绝对误差降为10~(-4),而且修正后的方法扩大了原数值解的收敛域.对于Benjamin-Bona-MahonyBurgers方程,利用带有辅助参数的变分迭代法将数值解的精度提高到10~(-3),对真解的逼近效果优于原始的变分迭代法.此数值方法也为其他强非线性孤立波微分方程的数值求解提供了方法和参考.     

优化模式物理参数的扩展四维变分同化方法

数值模拟的一个重要误差来源是模式物理参数,为提高模拟准确率,如何改进模式物理参数是亟需解决的问题.本文对经典四维变分同化技术进行了改进,提出了一种新的利用观测资料来同时优化模式初始场和物理参数的扩展四维变分同化方法,并以Ekman边界层模式和Lorenz模式为例进行了数值试验.结果表明,利用本文提出的新方法,通过对观测资料的变分同化,可以在实现对模式初始场进行优化的同时,纠正了模式物理参数中的误差,从而有效提高了模式的模拟准确率.该方法对于改进数值模式物理参数有着重要的促进意义.     

凑合反推法──流体力学变分原理建立的一条新途径

凑合反推法是刘商联系统方法[1]的进一步发展，应用这种方法可以方便地构造各种亚广义变分原理及广义变分原理，并可以消除临界变分现象．对于任何二维守恒型流体力学方程，作者推导得到了其广义变分通用公式．几个实例证明这种方法是有效的、简单的，并具有普遍的意义．     

基于变分方法的混沌系统参数估计

提出一种基于变分原理的估计混沌系统未知参数的方法,对以x= F(x,θ) 为控制方程的所有混沌系统具有普适性.首先将混沌系统方程引入到目标泛函中;接着利用变分原理导出了混沌系统的伴随方程和待辨识参数泛函梯度的通用公式;然后设计了估计混沌系统未知参数的算法;最后对典型的Lorenz混沌系统和超混沌Chen系统的未知参数进行了估计.数值仿真结果表明该方法是一种非常有效的估计混沌系统未知参数的方法. 关键词： 混沌系统 参数估计 变分方法 伴随方程     

基于二阶离散变分方法的非线性映射参数估计

提出一种估计非线性映射未知参数的二阶离散变分方法.首先针对非线性离散混沌系统, 利用变分方法导出了伴随方程和目标泛函梯度, 以此为基础利用二阶离散变分方法给出了二阶伴随方程和精确计算Hessian矩阵-向量乘积的显式表达式; 其次设计了估计非线性映射未知参数的新算法, 并以此对Hyperhenón映射和二维抛物映射中的未知参数进行了精确的估计. 数值仿真结果表明了该方法的有效性和优点. 关键词： 非线性映射 参数估计 二阶离散变分方法 伴随方程     

梯度计算的集合变分方案及其在大气Ekman层湍流系数反演中的应用

大气Ekman层湍流系数的准确计算对数值天气预报和污染物扩散计算有着重要的意义. 将集合计算和变分法结合起来, 提出了目标泛函梯度计算的集合变分方案, 并根据正演模式的线性化情况提出了两种计算流程. 利用这种集合变分梯度算法及两种流程对Ekman层湍流系数进行了反演试验, 结果表明这种算法实施简单、方便, 根据观测资料能够比较准确地反演湍流系数值. 关键词： 梯度计算 集合变分 湍流系数 反演     

二维线性与非线性光晶格中物质波孤立子的稳定性

利用变分法和数值计算方法研究了二维线性和非线性光晶格中二维玻色-爱因斯坦凝聚体系中物质波孤立子的存在及其稳定性. 利用定态变分原理及Vakhitov-Kolokolov判据总结了线性和非线性结合光晶格中几种参数组合下定态定域解的稳定性. 结果表明, 当存在二维非线性光晶格时, 在吸引和排斥相互作用的原子体系中均可以存在稳定的物质波孤立子. 另外, 利用含时变分法研究了线性和非线性光晶格中物质波孤立子随时间的传播特性, 使波包参数对时间的一阶导数等于零, 可以给出稳定状态对应的参数, 结论和定态变分法给出的结果一致. 最后用数值计算方法研究变分结果的正确性, 把变分结果作为初始条件代入Gross-Pitaevskii方程研究其随时间传播特征, 得到了稳定的传播过程, 所得到的结果和变分分析结果一致. 关键词： 线性非线性光晶格 玻色-爱因斯坦凝聚 孤立子 稳定性     

光纤台灯的包络面

由变分原理推导了重力场中弹性杆的形状方程.得到了竖直放置光纤开始弯曲时临界长度的数值结果.数值模拟给出了光纤末端形成的包络曲面,并讨论了它与光纤长度的关系.     

球型核壳量子点结构中二类激子向一类激子的转化

在有效质量近似下,采用变分方法研究了球型核壳量子点中激子的性质。数值计算了电子波函数随位置的变化关系以及临界势随球壳宽度和核半径的比值的变化关系。结果表明,当球壳宽度和核半径的比值为给定值时,存在一临界势,当局域势减小到临界势时,激子从二类激子转化为一类激子;当局域势为给定值时,存在一临界比值,当球壳宽度和核半径的比值减小到临界比值时,激子亦从二类激子转化为一类激子。该结果对研究球型核壳量子点中的电子结构有参考价值。     

Thermal explosion in a stirred medium

The critical conditions of a thermal explosion in a cylindrical chemical reactor with several symmetrically positioned agitators for creating a forced convection of the reaction mixture are examined. The analysis is carried out for an arbitrary Peclet number on the assumption that the flow of the fluid is laminar. The critical value of the thermal explosion parameter (Frank-Kamenetskii parameter) is determined as a function of the number of agitators, Peclet number, and the distances between the axes of the agitators and the reactor axis. In contrast to the previous studies, the dependence of the thermal explosion parameter on the rate of stirring of the medium is considered. Calculations show that, as the number of agitators and mixing rate increases, the thermal explosion probability decreases, so that the critical value of the thermal explosion parameter can be several times greater than its classical value. It is demonstrated that, depending on the values of the parameters, the problem can have four stationary solutions, two of which are stable.     

A New Principle to Determine the Variational Parameter for the Variational-Cumulant Expansion

The key point of the variational-cumulant expanhon is the determination of the variational parameter. In this paper, we present the improved mean-field hypothesis (IMFH) which is the bdse to determine the parameter. The new method derived from the IMFH shows advantage over previous methods. The critical temperature and some thermal dynamical functions for the Heisenberg model are calculated with the new method.     

Thermal explosion of a reactive mixture under conditions of oscillating ambient temperature

The problem of the thermal explosion of a finite volume of a reactive material in a medium with harmonically oscillating temperature is solved in the classical formulation. A kind of resonance is shown to arise when the oscillation period is commensurate with the adiabatic induction period of thermal explosion at the mean ambient temperature. At both high and very low oscillation frequencies, the critical condition parameter and induction period are only slightly affected by ambient temperature oscillations. By contrast, at moderately low frequencies, even small-amplitude oscillations of ambient temperature can strongly influence the critical condition and, especially, induction period of thermal explosion.     

The structure of the thermal explosion limit with variable pre-exponential factor

An approximate critical explosion parameter, which separates the slow and explosive reactions is determined with respect to the initial data, using the asymptotic technique to analyze a one step reaction mechanism with temperature dependent pre-exponential factor. We obtained and analysed the effect of the variable pre-exponential factor on the critical temperature, critical parameter for thermal explosion and the first order approximation of the slow curve. By introducing a new coordinate and using the Method of Invariant Manifold (MIM), we obtained corrections on the critical temperature. The results obtained are not only general; they also reveal some salient properties of the system hitherto unknown.     

Thermal explosion in a batch reactor charged with a liquid–solid heterogeneous system

The characteristics of a thermal explosion in an ideal mixing batch reactor charged with a liquid–solid heterogeneous system are studied. The reactor initially contains both phases. The solid reagent dissolves and reacts in the liquid phase. A strong dependence of the critical value of the Semenov parameter on the dimensionless time of complete dissolution of the solid reagent is established. It is shown that, at short times of complete dissolution, the critical value of the Semenov parameter is practically independent on this time, and the thermal explosion occurs as in a homogeneous system, according to Semenov theory. The heterogeneous properties of the reaction system manifest themselves only at long times of complete dissolution.     

On the Thermal Explosion Theory of a Heterogeneous Liquid–Liquid System

In this paper, the regularities of a thermal explosion of a heterogeneous system consisting of two immiscible liquids have been studied. Each phase is a solution of A and B reagents. Reagent B is extracted into a solution of reagent A, where the bimolecular exothermic reaction A + B → Products takes place. It has been shown that an exothermic reaction (combustion regime) continues to proceed in the system at high mass-exchange rates between phases after a thermal explosion. As a result, the maximal temperature may significantly exceed the temperature of the thermal explosion. The critical value of the Semenov parameter decreases with an increase in the mass-exchange rate between phases. In the limited range of values of the distribution coefficient of reagent B between phases, the increase of this coefficient is also accompanied by a decrease in the critical value of the Semenov parameter. The concentration of reagent B in the initial phase decreases monotonically due to its extraction into another phase. However, the equilibrium of the extraction of reagent B can shift, due to the temperature dependence of the distribution coefficient during the reaction. Thus, the time dependence of the concentration of reagent B on may be more complex and can pass through a minimum.     

Bright soliton in an expulsive potential in a dipolar condensate

We study the stability of matter-wave bright solitons for a dipolar Bose-Einstein condensate in a harmonic potential which is transversely attractive and longitudinally expulsive. We map the regimes of stability as a function of dipole-dipole interaction strength using sech-based variational ansatz. The critical points for collapse and explosion are presented.     

The Variational Cumulant Expansion Study of Spatially Anisotropic XY Model

With the variational-cumulant expansion (VCE) method, the critical temperature and spontaneous magnetization of the spatially anisotropic XY model are calculated up to the fourth order. The variational parameter is determined by the complete variational method and the results are compared with both the Monte Carlo results and the self-consistent harmonic approximation (SCHA) results. It is shown that the VCE method can give a better result than the SCHA method when the coupling between adjacent layers is small.     

Dual Variational Form of the Model of Thermal Explosion in a Quiescent Medium with Temperature-Dependent Thermal Conductivity

Based on the dual variational formulation of the nonlinear stationary heat conduction problem, a mathematical model is constructed that describes the temperature state of a stationary medium in a volume of an arbitrary shape. The thermal conductivity of the medium and the volumetric energy release depend on the temperature. This model is applied to analyzing the conditions of thermal explosion in an infinitely long circular cylinder for exponential temperature dependences of the above parameters. In the particular case of constant thermal conductivity of this cylinder, a comparison with the published results is carried out.     

Analytical Holographic Superconductor with Backreaction Using AdS 3/CFT 2

The holographic model for a two-dimensional superconductor has been investigated by considering the three-dimensional gravity in the bulk. To find the critical temperature, we used the Sturm–Liouville variational method. Where as the same method is applied for calculating the condensation of the dual operators on the boundary. We included the back reactions on the metric by a combination of the perturbation method of the fields with respect to the small parameter and then applying the variational integrals on the resulting equations of the motion. The critical temperature has been successfully obtained on the backreaction effects, and we showed that it dropped with a rise in the backreaction of the fields, and it makes the condensation harder. We can use our analytical results to support the numerical data which was reported previously.