基于卫星高度计海面高度异常资料获取潮汐调和常数方法及应用

利用25年间南海区域的卫星星下观测点数据处理分析南海潮汐问题.首先,选取了大于100个观测时间的4325个观测点,根据调和分析法计算了21个分潮的潮汐调和常数,并与潮汐验潮站的历史数据进行对比,验证分析方法的可靠性;在此结果的基础上,选取特定轨道,拟合各分潮沿轨道振幅和迟角曲线,分离正压潮和内潮,确定了拟合的最佳次数;根据轨道的数值插值到全南海海域,作出南海各主要分潮的同潮图,并进行误差检验.     

卫星高度计资料反演潮汐调和常数研究与应用

海洋潮汐是海洋科学研究中的经典问题,其研究对于军事,交通,环境,经济等都有着重要作用.主要内容包括:选择时间序列长于2.97年的高度计数据,使用带约束的最小二乘法求解潮汐调和常数;分别使用上升轨道和下降轨道,验潮站与邻近星下点进行对比,分析了K1潮的误差;将卫星星下点轨迹划分为2条轨道,对比了傅里叶级数和多项式级数拟合正压潮的方法,确定6阶傅里叶级数拟合和9阶多项式拟合将得到最优的绝均差;使用局部加权回归方法(LOWESS),并采取鲁棒优化策略,对各主要分潮调和常数进行插值,确定参与回归个数为25%时均方根误差最小;将插值后的潮汐调和常数与验潮站进行对比,结果表明,傅里叶级数拟合方法可与91个测站进行匹配,而多项式拟合仅能和38个验潮站进行匹配;根据两种插值方法求解的潮汐调和常数,绘制了K1潮的同潮图.     

利用边界元法求解一类重调和方程

边界元法(BEM)和多重互易法(MRM)相结合求解一类重调和方程.通过重调和基本解序列给出的MRM-方法和BEM, 推导出该类问题的MRM-边界变分方程, 用边界元法求解该变分方程, 从而得到重调和方程的近似解, 并给出了解的存在唯一性证明.通过数值算例说明了MRM-方法具有收敛速度快、计算精度高, 易编程等优点, 为使用边界元法数值求解重调和方程提供了方法和理论依据.适合于工程中的实际运算.     

数据同化中的伴随方法的有关问题的研究

关于伴随方法应用中只能利用模式的伴随的观点,被认为是有疑问的．所作的数值模拟实验表明,对于潮波模型而言,方程的伴随能够得到与模式的伴随同样的结果:调和常数的实测值与模拟值的振幅差的绝对值的平均小于5.0 cm,迟角差的绝对值的平均小于5.0°．这些结果都能够体现渤、黄海M2分潮的基本特征．作为对比,也利用前人的方法对渤、黄海的M2分潮潮波进行了数值模拟,首先借助于历史资料和观测资料得到开边界的初始猜测,然后对开边界的初始猜测值进行调整,以得到与高度计资料之差尽可能小的模拟结果．但由于开边界的值共有72个,究竟有哪些值需要调整,需要如何调整,只有经过不断的调试,才能部分地解决这些问题．工作量大且很难得到令人满意的结果．该文实现了确定开边界条件的自动化过程,这与前人的方法相比,有无可比拟的优势．需要特别强调的是如果利用方程的伴随,可以避免繁琐而冗长的数学推导．因而说明方程的伴随也应该引起足够的重视．     

局部资料下变分同化方法的稳定性研究

对于简化的一维扩散方程,在局部观测资料下,研究变分同化方法的稳定性.在变分同化中结合正则化方法,选择合适的正则化参数和稳定泛函,对预报模式进行修正,通过对预报精度进行先验估计,证明了该方法对于一维扩散方程的解的稳定性.修正补充相关计算结果,最后举出一个反例说明稳定性泛函的选取对于改进的变分同化方法实施的重要性.     

2018年中国研究生数学建模竞赛D题综述

介绍了2018年中国研究生数学建模竞赛D题"基于卫星高度计海面高度异常资料获取潮汐调和常数方法及应用"的命题背景和目的,分析了本赛题的建模及求解思路,对评阅中发现的问题进行了综述,最后提出了本赛题还需继续思考的问题.     

曲率障碍下四阶变分不等式的交替方向乘子法北大核心CSCD

对于重调和算子和曲率障碍表示的变分不等式,提出了自适应交替方向乘子数值解法(SADMM).对问题引入一个辅助变量表示曲率函数的增广Lagrange函数,导出一个约束极小值问题,并且该问题等价于一个鞍点问题.然后采用交替方向乘子法(ADMM)求解这个鞍点问题.通过采用平衡原理和迭代函数,得到了自动调整罚参数的自适应法则,从而提高了计算效率.证明了该方法的收敛性,并给出了利用迭代函数近似罚参数的具体方法.最后,用数值计算结果验证了该方法的有效性.     

关于平面调和映射的Bloch常数的估计

考虑了拟正则调和映射和开的平面调和映射的Bloch常数, 得到了较好的结果,
所得结果推广了陈怀惠等及Grigoryan的结果.     

类KW2[a,b]基于Hermite信息的最佳求积公式

找到了下述意义下的最佳求积公式: 对于在给定区间上二阶导数的模不超过给定常数的函数, 如果已知它在该区间上的若干点上的函数值和导数值, 则用该求积公式计算它的积分的近似值可以使最大可能的误差达到最小. 也给出了相应的最佳插值方法, 并用它来导出上述最佳求积公式. 同时, 还通过理论分析和随机数值试验把它和开型复合校正梯形公式做了比较.     

计算最小奇异组的一个精化调和 Lanczos双对角化方法

在很多实际应用中需要计算大规模矩阵的若干个最小奇异组.调和投影方法是计算内部特征对的常用方法,其原理可用于求解大规模奇异值分解问题.本文证明了,当投影空间足够好时,该方法得到的近似奇异值收敛,但近似奇异向量可能收敛很慢甚至不收敛.根据第二作者近年来提出的精化投影方法的原理,本文提出一种精化的调和Lanczos双对角化方法,证明了它的收敛性.然后将该方法与Sorensen提出的隐式重新启动技术相结合,开发出隐式重新启动的调和Lanczos双对角化算法(IRHLB)和隐式重新启动的精化调和Lanczos双对角化算法(IRRHLB).位移的合理选取是算法成功的关键之一,本文对精化算法提出了一种新的位移策略,称之为"精化调和位移".理论分析表明,精化调和位移比IRHLB中所用的调和位移要好,且可以廉价可靠地计算出来.数值实验表明,IRRHLB比IRHLB要显著优越,而且比目前常用的隐式重新启动的Lanczos双对角化方法(IRLB)和精化算法IRRLB更有效.     

Internal Kelvin waves in a two-layer liquid model

The subinertial internal Kelvin wave solutions of a linearized system of the ocean dynamics equations for a semi-infinite two-layer f-plane model basin of constant depth bordering a straight, vertical coast are imposed. A rigid lid surface condition and no-slip wall boundary condition are imposed. Some trapped wave equations are presented and approximate solutions using an asymptotic method are constructed. In the absence of bottom friction, the solution consists of a frictionally modified Kelvin wave and a vertical viscous boundary layer. With a no-slip bottom boundary condition, the solution consists of a modified Kelvin wave, two vertical viscous boundary layers, and a large cross-section scale component. The numerical solutions for Kelvin waves are obtained for model parameters that take account of a joint effect of lateral viscosity, bottom friction, and friction between the layers.     

Stokes波高阶谐波系数递推的数值计算*

本文用W.H.Hui提出的方法,在半物理平面内重新表述了Stokes波的数学模型和边界条件,提出了两种更有效的数值计算方法来获得Stokes波高阶谐波系数,并可递推至无穷.通过小参数转换,重新得到了Cokelet(1977)的波速和半波高的摄动展开式.     

A 3D hydrodynamic numerical model of the Río de la Plata and Montevideo’s coastal zone

The 3D hydrodynamic numerical model MOHID was applied in the Río de la Plata and Montevideo coastal zone in order to represent the main dynamics and to study its complex circulation pattern. The hydrodynamic model was calibrated and validated considering the following main forces: fresh water flow, astronomical and meteorological tides in the oceanic boundary, and wind acting on the water surface. A series of water levels measured at six coastal stations and vertical profiles of current velocity measured at four different locations in the estuarine zone of the Río de la Plata were used for calibrating and validating the hydrodynamic model. The calibration process was carried out in two steps. First the astronomical waves propagation was calibrated comparing harmonic constants of observed and computed sea surface elevation data. Next, both the astronomical and meteorological wave propagation was calibrated. Direct comparison of scatter plot and root-mean square errors of model results and field data were used when evaluating the calibration quality. The calibrated model shows good agreement with the measured water surface level in the entire domain with mean error values being minor than 20% of the measured data and correlation factors higher than 0.74. Also, the intensity and velocity direction observed in the currents data are well represented by the model in both bottom and surface levels with errors similar to 30% of the currents data components. Using the 3D calibrated model the bottom and surface residual circulation for a four month period of time was analyzed.     

Local energy decay of solutions of linearized shallow-water equations

We prove the local decay of the energy of the solution to a mixed initial boundary value problem for the linearized shallow-water equations with constant coefficients, where the domain is a half-plane, a certain dissipative boundary condition is prescribed and the initial data have compact support contained in the open half-plane.     

Elastodynamic Contact Problems for Interface Cracks under Harmonic Loading

2-D fracture dynamics' problems for elastic bimaterials with cracks located at the bonding interface under the oblique time harmonic wave are considered in the study. The system of boundary integral equations for displacements and tractions is derived from Somigliana identity taking the contact interaction of the opposite crack faces into account. For the numerical solution the collocation method with piecewise constant approximation is used. The numerical results are obtained for various values of the angle of the wave incidence and the wave frequency taking the friction effects into account. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

椭圆边界上的自然积分算子及各向异性外问题的耦合算法

1.引 言为求解微分方程的外边值问题常需要引进人工边界（见[1-4]）,对人工边界外部区域作自然边界归化得到的自然积分方程即Dirichlet-Neumann映射,正是人工边界上的准确的边界条件（见[2-6]）,这是一类非局部边界条件.自然积分算子即Dirichlet-Neumann算子,     

Numerical simulation of the turbulent boundary layer equations via a Runge-Kutta algorithm

This paper addresses a hybrid computational procedure for the step-by-step calculation of momentum transfer in turbulent boundary layer flows along flat plates. The proposed procedure relies on a modified method of lines wherein transversal discretizations are carried out by a “control volume” being infinitesimal in the streamwise direction and finite in the transversal direction of the fluid flow. Using mixing length theory and coarse intervals in the transversal direction, the resulting system of ordinary differential equations of first order may be readily integrated on a personal computer utilizing a fourth-order Runge-Kutta algorithm. In general, a maximum number of sixteen lines is necessary at the trailing edge of the flat plate for a typical calculation. As a consequence, computing time and storage for each run were very small when compared to other finite-difference methods. Furthermore, to validate the hybrid procedure involving the method of lines and control volumes (MOLCV), comparisons with experimental data have been done in terms of both velocity distributions and local skin friction coefficients. For all cases tested, the proposed methodology predicts the growth of the boundary layer of gases correctly.     

Solvability of elliptic systems with square integrable boundary data

We consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L ₂ Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when A is either Hermitean, block or constant. Our methods apply to more general systems of partial differential equations and as an example we prove perturbation results for boundary value problems for differential forms.     

A harmonic polynomial method with a regularization strategy for the boundary value problems of Laplace’s equation

This paper concerns a boundary value problem of Laplace’s equation, which is solved by determining the unknown coefficients in the expansion of harmonic polynomials. A regularization method is proposed to tackle the resulting ill-posed linear system. The stability and convergence results are provided and a validating numerical experiment is presented.