Parareal Algorithms Applied to Stochastic Differential Equations with Conserved Quantities

In this paper, we couple the parareal algorithm with projection methods of the trajectory on a specific manifold, defined by the preservation of some conserved quantities of stochastic differential equations. First, projection methods are introduced as the coarse and fine propagators. Second, we apply the projection methods for systems with conserved quantities in the correction step of original parareal algorithm. Finally, three numerical experiments are performed by different kinds of algorithms to show the property of convergence in iteration, and preservation in conserved quantities of model systems.     

含源项浅水波方程的稳态解及其计算方法

Consider the shallow fluid flow in a channel with wavy bottom, which is governed by the following equations: ht (hu)x=0, (hu)t (hu^2 gh^2/2)x=-ghB‘(x),(1)     

On the Expected Number of Zeros of Nonlinear Equations

This paper investigates the expected number of complex roots of nonlinear equations. Those equations are assumed to be analytic, and to belong to certain inner product spaces. Those spaces are then endowed with the Gaussian probability distribution. The root count on a given domain is proved to be ‘additive’ with respect to a product operation of functional spaces. This allows one to deduce a general theorem relating the expected number of roots for unmixed and mixed systems. Examples of root counts for equations that are not polynomials, nor exponential sums are given at the end.     

Model Equations and Traveling Wave Solutions for Shallow-Water Waves with the Coriolis Effect

In the present study, we start by formally deriving the simplified phenomenological models of long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the Earth’s rotation. These new model equations are analogous to the Green–Naghdi equations, the first-order approximations of the KdV-, or BBM type, respectively. We then justify rigorously that in the long-wave limit, unidirectional solutions of a class of KdV- or BBM type are well approximated by the solutions of the Camassa–Holm equation in a rotating setting. The modeling and analysis of those mathematical models then illustrate that the Coriolis forcing in the propagation of shallow-water waves can not be neglected. Indeed, the CH-approximation with the Coriolis effect captures stronger nonlinear effects than the nonlinear dispersive rotational KdV type. Furthermore, we demonstrate nonexistence of the Camassa–Holm-type peaked solution and classify various localized traveling wave solutions to the Camassa–Holm equation with the Coriolis effect depending on the range of the rotation parameter.

     

转动相对论系统的Lie对称性和守恒量

研究转动相对论性完整与非完整力学系统的Lie对称性和守恒量.定义转动相对论力学系统的无限小变换生成元,利用微分方程在无限小变换下的不变性,建立转动相对论性力学系统的Lie对称确定方程,得到结构方程和守恒量的形式,并给出应用实例.     

On the Shallow Water Equations at Low Reynolds Number

The vertial averages of the incompressible Navier-Stokes equaitons are studied from the point of view of numerical analysis:existence of solution and converagence of algorithms. Three formulations are analysed; existence theorems are obtained when the Reynolds number is small. Convergene of a time implict algorithm is shown, while discretization in space is achived with the finite element method Résumé On étudie, due vue de l'analyse numérique, le problé obtenu en Prenant la moyenne verticale des équations de Navier-stokes: on s'intéresse á l'existence de solutions et á la convergence d'algorithmes. Trois formulations sont analysées; on déor$eacute;mes d'existence pour de faibles valeurs du nombrede Reynolds. On prouve la convergence d'un schéma implicite en temps, tandis que la discr$eacute;tisation en espace est effectuée par éléments finis     

Regularized Equations for Numerical Simulation of Flows in the Two-Layer Shallow Water Approximation

Regularized equations describing hydrodynamic flows in the two-layer shallow water approximation are constructed. A conditionally stable finite-difference scheme based on the finitevolume method is proposed for the numerical solution of these equations. The scheme is tested using several well-known one-dimensional benchmark problems, including Riemann problems.     

二阶非完整力学系统的Lie对称性与守恒量

研究二阶非完整力学系统的Lie对称与守恒量.首先利用系统运动微分方程在无限小变换下的不变性建立Lie对称的确定方程和限制方程,得到Lie对称的结构方程和守恒量;其次研究上述问题的逆问题;最后举例说明结果的应用.     

变质量完整力学系统的Lie对称与守恒量

研究变质量完整系统的Lie对称和守恒量。利用常微分方程在无限小变换下的不变性建立系统Lie对称的确定方程。给出结构方程和守恒量。举例说明结果的应用。     

关于有限域上一类方程解的个数

Let Fq be a finite field with q = pf elements,where p is an odd prime.Let N(a1x12 + ···+anxn2 = bx1 ···xs) denote the number of solutions(x1,...,xn) of the equation a1x12 +···+ anxn2 = bx1 ···xs in Fnq,where n 5,s n,and ai ∈ F*q,b ∈ F*q.In this paper,we solve the problem which the present authors mentioned in an earlier paper,and obtain a reduction formula for the number of solutions of equation a1x21 + ··· + anxn2 = bx1 ···xs,where n 5,3 ≤ s n,under a certain restriction on coefficients.We also obtain an explicit formula for the number of solutions of equation a1x21 + ··· + anxn2 = bx1 ···xn-1 in Fqn under a restriction on n and q.     

一类不定方程组的解的个数

本文将证明以下结论:设m为正整数,当(a,c,δ)取(m,m+1,-1),(m,m+ 2,-2),(m,m+4,-4),或者(m+2,m,2)时,联立的不定方程组■的正整数解(x,y,z)的个数不超过1。     

Exact Step-Like Solutions of One-Dimensional Shallow-Water Equations over a Sloping Bottom

Mathematical Notes -     

时滞Lagrange系统的Lie对称性与守恒量研究

研究了位形间中含单时滞参数的非保守力学系统的Lie对称性和守恒量。首先,利用含时滞的动力学Hamilton原理,建立了含时滞的非保守系统的分段Lagrange运动方程;其次,利用微分方程容许Lie群理论,得到系统的Lie对称确定方程;然后,根据对称性与守恒量之间的关系,通过构造结构方程,得到含时滞的非保守系统的Lie定理;最后,给出了两个具体的算例说明了方法的应用。结果表明:时滞参数的存在使非保守系统的Lagrange方程呈现分段特性,相应的Lie对称性确定方程的个数应是自由度数目的2倍,这对生成元函数提出了更高的限制,同时,守恒量呈现依赖速度项的分段表达。     

汽车车体振动系统的对称性与守恒量研究

用Lie群方法研究汽车车体振动系统的对称性,寻找其存在的守恒量.以汽车车体做上下垂直振动和绕其质心的前后俯仰振动,采用Lagrange函数的方法,构建汽车车体振动系统.以此系统为对象,引入Lie群方法,给出该振动系统的Noether对称性理论与Lie对称性理论;由此推导该汽车系统存在的Noether对称性与Lie对称性,并得到系统相应的的守恒量.该方法对车体振动问题提出了新的对称性解法,同时扩大了Lie群方法的应用范围.     

关于有限域上一类方程零点个数的估计

By establishing the connection between graph colouring and the solution of some equation systems in finite fields, we obtain some formulas to the number of solutions of some equation systems in finite fields, in terms of chromatic polynomial of a graph.     

Common Conserved Densities AKNS Hierarchies of Equations

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非完整非保守力学系统在相空间的Lie对称性与守恒量

在相空间引入无限小变换,研究非完整非保守力学系统运动微分方程的不变性和守恒量。建立Lie对称确定方程,得到Lie对称的结构方程和守恒量形式,并举例说明结果的应用。