Element-free Galerkin method for a kind of KdV equation

The present paper deals with the numerical solution of the third-order nonlinear KdV equation using the element-free Galerkin (EFG) method which is based on the moving least-squares approximation. A variational method is used to obtain discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for KdV equations needs only scattered nodes instead of meshing the domain of the problem. It does not require any element connectivity and does not suffer much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for the KdV equation is investigated by two numerical examples in this paper.     

Numerical solution of the Fokker-Planck equation with variable coefficients

We study the numerical solution of the Fokker-Planck equation. This equation gives a good approximation to the radiative transport equation when scattering is peaked sharply in the forward direction which is the case for light propagation in tissues, for example. We derive first the numerical solution for the problem with constant coefficients. This numerical solution is constructed as an expansion in plane wave solutions. Then we extend that result to take into account coefficients that vary spatially. This extension leads to a coupled system of initial and final value problems. We solve this system iteratively. Numerical results show the utility of this method.     

一类相对论性非球谐振子系统的束缚态

给出了具有形式为12r²+A2r²的非球谐振子型标量势和矢量势 的相对论系 统在两种势相等的条件下三维Klein-Gordon方程，二维和三维Dirac方程的s波束缚态解. 关键词： 三维非球谐振子势 Klein-Gordon方程 Dirac方程 束缚态     

强阻尼广义sine-Gordon方程特征问题的变分迭代法

研究了在数学、力学中广泛出现的一类非线性强阻尼广义sine-Gordon扰动微分方程问题. 首先, 引入行波变换, 求出退化方程的精确解. 再构造一个泛函, 创建了一个变分迭代算法, 最后, 求出原非线性强阻尼广义sine-Gordon扰动微分方程问题的近似行波解析解. 用变分迭代法可得到的各次近似解, 具有便于求解、精度高等特点. 求得的近似解析解弥补了单纯用数值方法的模拟解的不足.     

广义Zakharov-Kuznetsov方程的多辛Preissmann格式

广义Zakharov-Kuznetsov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,基于Hamilton空间体系的多辛理论研究了广义Zakharov-Kuznetsov方程的数值解法,讨论了利用Preissmann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、局部能量守恒律.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.     

广义Zakharov-Kuznetsov方程的多辛Preissmann格式

广义Zakharov-Kuznetsov 方程作为一类重要的非线性方程有着许广泛的应 用前景,基于Hamilton 空间体系的多辛理论研究了广义Zakharov-Kuznetsov方程的数值 解法,讨论了利用Preissmann 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、局部能量守恒律. 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.     

用格子Boltzmann模型模拟垂直平板间的热对流

引入一个新的能量分布函数，利用该能量分布函数与粒子速度分布函数耦合来求解一个热流场. 因而，这一能量分布函数与粒子速度分布函数和Boltzmann方程构成了一个新的格子Boltzmann模型. 这一模型满足质量、动量和能量守恒的准则. 用该模型对垂直平板间的狭缝热对流进行了数值模拟，数值结果表明，在Prandtl数为1，Grashof数在1.3×10²—1×10⁶之间时，流场将出现多个旋涡结构的流型. 得出了与Lee相一致的结论. 关键词： 能量分布函数 Boltzmann方程 热对流     

Multiple lump molecules and interaction solutions of the Kadomtsev–Petviashvili I equation

In this paper, a modified version of the solution in form of a Gramian formula is employed to investigate a new type of multiple lump molecule solution of the Kadomtsev–Petviashvili I equation. The high-order multiple lump molecules consisting of M N-lump molecules are constructed by means of the Mth-order determinant and the non-homogeneous polynomial in the degree of 2N. The interaction solutions describing P line solitons radiating P of the M N-lump molecules are constructed. The dynamic behaviors of some specific solutions are analyzed through numerical simulation. All the results will enrich our understanding of the multiple lump waves of the Kadomtsev–Petviashvili I equation.     

希罗喷泉动力学分析

本文构建了一个希罗喷泉系统,利用修正后的非定常伯努利方程及连续性方程,在层流与非定常的基本假设下,解出其动力学方程;使用Mathematica软件并结合实验参数进行数值求解,推导了喷射高度随时间的理论变化关系,并探讨了喷射瓶的相对放置高度、横截面积、喷射管道直径以及喷口高度等参量对喷泉高度的影响.同时运用Tracker软件追踪喷泉高度,建立了空气弹簧的模型,完成了理论与实验的相互验证.     

解三维扩散方程的积分内插法格式

研究三维扩散方程的数值模拟.在非正规六面体网格上,使用积分内插法建立扩散方程差分格式,涉及到27个相邻网格,适用于大变形网格上带间断系数的拟线性扩散方程的计算.叙述差分格式的建立,推导通量流和网格顶点温度的计算公式,给出了数值试验结果.     

基于变分原理的三维扩散方程离散化格式

研究三维扩散方程的数值模拟.在非正规六面体网格上,用变分原理建立扩散流形式的差分格式.将扩散流通量作为未知函数,求其泛函极值,并与原函数温度联立求解.详细推导了计算公式,并给出了数值试验结果.     

势问题的径向基函数——局部边界积分方程方法

将基于径向基函数构造的具有插值特性的近似函数和局部边界积分方程方法相结合，建立了求解势问题的径向基函数——局部边界积分方程方法，推导了相应离散方程.与其他边界积分方程的无网格方法相比，本文方法具有数值实现过程简单、计算量小、精度高的优点，并可直接施加边界条件.最后通过算例说明了该方法的有效性. 关键词： 径向基函数 无网格方法 局部边界积分方程 势问题     

Fast sweeping method for the factored eikonal equation

We develop a fast sweeping method for the factored eikonal equation. By decomposing the solution of a general eikonal equation as the product of two factors: the first factor is the solution to a simple eikonal equation (such as distance) or a previously computed solution to an approximate eikonal equation. The second factor is a necessary modification/correction. Appropriate discretization and a fast sweeping strategy are designed for the equation of the correction part. The key idea is to enforce the causality of the original eikonal equation during the Gauss–Seidel iterations. Using extensive numerical examples we demonstrate that (1) the convergence behavior of the fast sweeping method for the factored eikonal equation is the same as for the original eikonal equation, i.e., the number of iterations for the Gauss–Seidel iterations is independent of the mesh size, (2) the numerical solution from the factored eikonal equation is more accurate than the numerical solution directly computed from the original eikonal equation, especially for point sources.     

ISSDE:基于第一性原理随机微分方程的碰撞等离子体隐式模拟

开发一种适用于求解含库仑碰撞等离子体随机微分方程（SDE）的第一性原理隐式模拟程序——Implicit Stratonovich Stochastic Differential Equations （ISSDE）.该程序基于Fokker-Planck （FP）方程与Stratonovich SDE的等价性理论，通过精确求解Stratonovich SDE达到对FP方程的高效第一性原理计算的目标.ISSDE采用隐式格式保证求解的数值稳定性，同时可以保证粒子碰撞过程的能量守恒.ISSDE基于C++语言开发，具有标准接口和灵活的可扩展模组.通过模拟电子束在非磁化和磁化致密等离子体中的慢化过程验证ISSDE的正确性，并展示该程序在碰撞等离子体模拟中的应用.     

Numerical resolution of the modified Langevin equation using a differential expression: Application to the Jiles magnetostriction law of approach

The Langevin equation is classically used to model the anhysteretic magnetization curve. A modified version of this equation has been introduced by Jiles to take into account the effects of magnetostriction on the anhysteretic magnetization behavior when a ferromagnetic material undergoes mechanical stresses. The numerical resolution of the modified Langevin equation is usually performed with a root-finding algorithm. In this paper, a differential form of the modified Langevin equation is proposed, allowing a faster numerical resolution.     

Adomian Decomposition Method for Nonlinear Differential-Difference Equation

Adomian decomposition method is applied to find the analytical and numerical solutions for the discretized mKdV equation. A numerical scheme is proposed to solve the long-time behavior of the discretized mKdV equation. The procedure presented here can be used to solve other differential-difference equations.     

Quantum Discord in Two-Qubit System Constructed from the Yang–Baxter Equation

Quantum correlations among parts of a composite quantum system are a fundamental resource for several applications in quantum information. In general, quantum discord can measure quantum correlations. In that way, we investigate the quantum discord of the two-qubit system constructed from the Yang-Baxter Equation. The density matrix of this system is generated through the unitary Yang-Baxter matrix R. The analytical expression and numerical result of quantum discord and geometric measure of quantum discord are obtained for the Yang-Baxter system. These results show that quantum discord and geometric measure of quantum discord are only connect with the parameter θ, which is the important spectral parameter in Yang Baxter equation.     

Hojman's theorem of the third-order ordinary differential equation

This paper extends Hojman's conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results.     

托卡马克二维局部模的第三类气球理论

采用二维气球表象,在已有第一类和第二类气球理论的基础上,构造了第三类气球理论。与第一类气球理论相比,扩展了后者因受可解性条件限制的适用范围;与第二类气球理论相比,因容纳了“坏曲率”效应而获得具有更高增长率的解,故与采用初值数值模拟方法所得的结果更具可比性。本文以流体型离子温度梯度模为例,用第三类气球理论求解了二维本征值问题,计算了所选剖面参数下的本征频率和模式结构,并与前两种理论所得的结果进行了比较和讨论;也讨论了该理论对实现托卡马克高约束的意义。     

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The ballooning theory of the third kind is constructed in parallel to the ballooning theory of the first and the second kind by making use of the 2-D ballooning representation. Compared to the ballooning theory of the first kind, the restriction to applicability due to solvability condition is found greatly relaxed to wider parameter domain; and compared to the ballooning theory of the second kind, the solution with higher growth rate is obtained because the destabilization from “bad curvature” is well included within the framework of the theory, which in turn, makes the numerical results more comparable with those from initial value simulations. The fluid model equation for ion temperature gradient mode in toroidal coordinate is adopted to calculate growth rate and eigen frequency as well for selected profile parameters. The results are presented graphically, compared and discussed with the two previous ballooning theories; also discussed is the implication with regard to improved confinement in tokamaks.