Generalized characteristic interface conditions for high-order multi-block computation

In practical fluid computation with structured grids around complex geometries, singular points with metric discontinuity can frequently be found. Generally, the grid singularities may cause numerical oscillations when some high-order finite difference scheme is applied. Recently, an excellent theory has been proposed which solves the above singular problem by block decomposition along the singular surface and by imposition of the characteristic interface conditions (CIC) on the block interface. However, the original theory has constraints on the mathematical treatment of the block interface, and therefore prevents numerical flexibility from a practical point of view. In this article, in order to extend the functions of the original CIC, we propose the generalized characteristic interface conditions (GCIC). Proper numerical test analysis is conducted to validate the performance of the GCIC, and as a practical application, multi-block computation is performed with the GCIC applied to complex geometry.     

Multi-wavelength laser active coherent combination

A seed laser oscillating at different frequencies is proved to have the potential to mitigate the stimulated Brillouin scattering(SBS) effect in a fiber amplifier,which may increase the emission power of a coherent beam combination(CBC) system greatly.In this study,a basic mathematical model describing the multi-wavelength CBC is proposed on the fundamentals of CBC.A useful method for estimating the combination effect and analysing the feasibility and the validity of the multi-wavelength coherent combination is provided.In the numerical analysis,accordant results with four-wavelength four-channel CBC experiments are obtained.Through calculations of some examples with certain spectra,the unanticipated excellent combination effect with a few frequencies involved is explained,and the dependence of the combination effect on the variance of the amplifier chain length and the channel number is clarified.     

三元结构双反射平行分束偏光镜设计

为了节省冰洲石晶体材料,并实现两束偏振光的平行输出,采用冰洲石与光学玻璃组合的方法,给出一种双反射形式平行分束偏光镜的新设计。该棱镜为ZBaF3玻璃中间加冰洲石晶体薄片的三元结构,采用溴代萘胶合;选择合适的结构角,令电矢量振动方向相互垂直的两束偏振光分别在前胶合面、后端面发生全反射,并垂直上端面平行出射。实验测试表明,棱镜的透射比高于80%,消光比优于10-3。该设计在节省冰洲石晶体前提下,保持了良好的性能。     

顾及高程差异的网络差分对流层误差内插模型

在GPS网络差分定位技术中,用户站与参考站间的高程差异会引起对流层误差改正数中存在系统性偏差,造成对流层改正精度降低,影响用户定位的效果。在常规两维线性组合模型的基础上,考虑高程因素影响,利用三维位置信息对内插模型参数进行约束,将对流程延迟改正数分为距离相关和高程相关两部分,通过卡尔曼滤波器分别估计,提出了一种能修正高程偏差的对流层三维线性组合模型。以重庆连续运行参考站系统中5个站作为研究对象,实验验证结果表明,在地形起伏较大的地区,三维线性组合模型可将对流层改正精度从20～40 cm提高到5～6 cm,相对于常规方法提高了4～6倍,消除了网络差分定位中高程差引起的对流层改正系统偏差。     

渗流耦合系统动边值问题特征分数步差分方法

对多层渗流方程耦合系统动边值问题,提出适合并行计算的一类特征分数步差分格式,利用区域变换、变分形式、粗细网格配套、乘积型叁二次插值、差分算子乘积交换性、高阶差分算子的分解、先验估计的理论和技巧,得到收敛性的最佳阶l2误差估计．该方法已成功地应用到多层油资源运移聚集的资源评估生产实际中,得到了很好的数值模拟结果．     

Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach

In this paper an efficient computational method based on extending the sensitivity approach(SA) is proposed to find an analytic exact solution of nonlinear differential difference equations.In this manner we avoid solving the nonlinear problem directly.By extension of sensitivity approach for differential difference equations(DDEs),the nonlinear original problem is transformed into infinite linear differential difference equations,which should be solved in a recursive manner.Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained.Numerical examples are employed to show the effectiveness of the proposed approach.     

Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation

We present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on the $L1$-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.     

High Accuracy Split-Step Finite Difference Method for Schrödinger-KdV Equations

In this article, two split-step finite difference methods for Schrödinger-KdV equations are formulated and investigated. The main features of our methods are based on:(i) The applications of split-step technique for Schrödingerlike equation in time. (ii) The utilizations of high-order finite difference method for KdV-like equation in spatial discretization. (iii) Our methods are of spectral-like accuracy in space and can be realized by fast Fourier transform efficiently. Numerical experiments are conducted to illustrate the efficiency and accuracy of our numerical methods.     

四色猜想的解析论证及其在地图绘制中的应用

依据欧拉定理,研究了边数差和着色数计算公式,对四色猜想进行了研究．借助四面体顶点数与面积数相等的原则、多面体边数不变的原则和多余理论,用边数差数学计算方法论证了四色猜想．用简单的数学公式和几何作图方法说明了四色猜想的合理性,为其提供了可靠的理论依据．用“三色包点”和“以面切体”的几何作图法,证明多面体和平面地图的着色数恒为4;非三色包点的图形,可以通过“以面切体”的方法转换成三色包点的图形;使用多余国家、多余边数的数学技巧代替计算机使用的不可避免性、可约性是合适的．理论分析及实例论证表明该方法简单可行．     

水槽涡的变化规律数值模拟

建立了不可压缩Navier-Stokes方程的Crank-Nicolson有限差分方法,数值模拟水槽晃动中流场及其涡流的数值变化规律。将数值解与解析解和前人的数值解进行比较,数值验证了不可压缩Naver-Stokes方程有限差分方法的有效性。通过数值模拟得到水槽在不同程度的倾斜激励晃动下流场及涡流的数值变换规律,当倾斜激励晃动的频率接近或远离共振频率时,水槽涡场的变化逐步由双涡变成单涡,再到不规则的涡场。当倾斜激励晃动的频率靠近共振频率ω_p=0.95ω_1附近时,水槽流场上部形成一个小涡,然后小涡扩大成整个水槽中的大涡,大涡下沉分裂成两个单涡,最后在底部消失;当倾斜激励晃动的频率在ω_p=0.75ω_1附近时,水槽底部形成一个小涡,然后扩大成大的单涡,最后在自由面消失;当倾斜激励晃动的频率在ω_p=0.55ω_1附近时,水槽底部出现小涡,然后扩大成大的单涡,大涡在自由面消失,继而出现不规则的大涡和不规则的小涡。