用离散速度方法计算浅水长波方程

用离散速度法计算浅水波方程，将空气动力学方程和浅水波方程作了比较，用Nadiga提出的近平衡流动方法模拟浅水波方程的连续和间断解。计算了一维的溃坝波问题和Thacker提出的连续解问题，结果与精确解作了比较，并且计算了水流跃过障碍物的问题。     

Lattice Boltzmann方法应用实例--一维溃坝波模拟

通过分析Ｌａｔｔｉｃｅ Ｂｏｌｔｚｍａｎｎ（ＬＢ）方法的多尺度方程和一维浅水波方程,建立了计算一有渠非恒定流的ＬＢ模型,对一维溃坝波动这一典型应用实例进行了模拟分析,结果与Ｐｒｅｉｓｓｍａｎｎ隐格式及解析解十分一致,说明将ＬＢ方法引入实际的一维明渠非恒定流计算是可行的。     

层体边界元法在二维层体断裂分析中的应用

在刚度矩阵法的基础上建立了用于进行二维多层体结构断裂分析的边界单元法（ＢＥＭＬＭ）由于ＢＥＭＬＭ的基本方程中已经包含了层体表面和裂纹缝面的边界条件，因而不需要对这些边界进行单元离散，从而其断裂分析可望有较好的精度通过与柯西积分方程法进行结合，算例表明ＢＥ ＭＬＭ是可靠并有效的     

求解浅水波方程的并行物理信息神经网络算法

双曲守恒律方程是一类比较特殊的偏微分方程,其数值求解方法的研究一直是一个热点问题,一个显著特性是即使初始条件是光滑的,其解也可能会发展成间断。浅水波方程作为非线性双曲守恒律方程,由于间断解的存在,其精确求解存在很大困难。针对浅水波方程数值求解问题,本文基于PINN(Physics informed neural networks)反问题网络结构构造新的网络,构造的网络结构包括两个并行的神经网络,其中一个网络与已知状态数据(熵稳定格式加密求出)相关,另一个网络与方程本身相关。利用已知速度数据结合浅水波方程本身求解未知水深,最终通过一些数值算例验证网络的可行性。结果表明,新的网络结构可用于浅水波方程求解,利用速度数据可以较为精确地推算出水深。     

Boussinesq方程的发展形式与Airy波理论差异性研究

应用势流理论，采用递推函数方法推导出一个新形式的Ｂｏｕｓｉｎｅｓｑ方程。通过对新方程的参数设置，可以讨论出Ｂｏｕｓｓｉｎｅｓｑ方程发展趋势和不同的发展形式。对浅水波动的描述方程，Ｂｏｕｓｓｉｎｅｓｑ方程的发展趋势为适用水深范围的拓展。拓展应用范围的大小则由其方程频散特征向Ａｉｒｙ波频散解逼近程度来决定。而Ｂｏｕｓｉｎｅｑ方程又不同于Ａｉｒｙ波，主要原因是Ｂｏｕｓｓｉｎｅｓｑ方程中含有线性频散项，Ａｉｒｙ波则只是长波首项近似，无线性频散项。其频散特征为精确的线性频散解。对实际水波传播而言，Ａｉｒｙ波理论的局限性是不言而喻的。     

Kupershmidt浅水波方程的线性化,相似约化和孤波解

利用未知函数变换方法，找到了Ｋｕｐｅｒｓｈｍｉｄｔ方程到Ｂｕｒｇｅｒｓ方程及热传导方程间的Ｂａｃｋｌｕｎｄ变换并借此给出了Ｋｕｐｅｒｘｈｍｉｄｔ方程四种杨似约化和一组孤波解。     

促进其线性频散特征另一种形式的Bousinesq方程

Ｂｏｕｓｉｎｅｓｑ方程能够用于模拟表面重力波传播过程中的折射、绕射、反射以及浅化，非线性作用等现象．用不同垂直积分方法所得到的二维Ｂｏｕｓｓｉｎｅｓｑ方程形式具有不同的线性频散特征．采用两个不同的水深层的水平速度变量组合，推导出一个新形式的Ｂｏｕｓｉｎｅｓｑ方程．通过对其参数的设置可得到精确的线性频散解Ｐａｄｅ近似４阶精度．其适用范围已由原来的浅水，向深水拓进．相速误差小于２％，其拓展适用范围可达到０８个波长水深．应用所得到的新型Ｂｏｕｓｉｎｅｓｑ方程，采用有限差分法，对经典工况进行了数值模拟，其计算结果表明，计算值与物模实验值吻合较好．这说明本文新形式的Ｂｏｕｓｓｉｎｅｓｑ方程对变水深非线性效应所产生的能量频散有着较为精确的描述     

促进其线性频散特征另一种形式的Bousinesq方程

Ｂｏｕｓｉｎｅｓｑ方程能够用于模拟表面重力波传播过程中的折射、绕射、反射以及浅化，非线性作用等现象．用不同垂直积分方法所得到的二维Ｂｏｕｓｓｉｎｅｓｑ方程形式具有不同的线性频散特征．采用两个不同的水深层的水平速度变量组合，推导出一个新形式的Ｂｏｕｓｉｎｅｓｑ方程．通过对其参数的设置可得到精确的线性频散解Ｐａｄｅ近似４阶精度．其适用范围已由原来的浅水，向深水拓进．相速误差小于２％，其拓展适用范围可达到０８个波长水深．应用所得到的新型Ｂｏｕｓｉｎｅｓｑ方程，采用有限差分法，对经典工况进行了数值模拟，其计算结果表明，计算值与物模实验值吻合较好．这说明本文新形式的Ｂｏｕｓｓｉｎｅｓｑ方程对变水深非线性效应所产生的能量频散有着较为精确的描述     

位移浅水内孤立波

研究两层浅水系统中的内孤立波，该系统由两层常密度不可压缩无黏性水组成。利用Lagrange坐标和Hamilton原理，推导了两层浅水系统的位移浅水内波方程，并进一步导出了两层浅水系统的位移内孤立波解。数值实验表明，位移内孤立波与经典的KdV内孤立波吻合很好，说明Lagrange坐标和Hamilton方法适用于内波分析，可以为构造内波分析的保辛方法提供一种途径。     

关于采用高阶Boussinesq方程求解波浪散射问题的注记

本文讨论了采用高阶Boussinesq方程模拟波浪散射时对基本速度变量位置的局部光滑处理方法。通过光滑局部基本速度变量的取值深度，减小其高阶导数项的量值、加快级数收敛速度进而改善模型方程求解深水波浪散射问题的能力。对于底部边界具有一阶导数不连续的情况，通过局部光滑．可以将基本速度变量取值深度尖角转化为圆角过渡，从而改善速度分布。对于其它任意变化的底部边界，为了减少高阶底坡导数项的影响，在曲率和高阶底坡导数项与斜率具有相同量级的情况下亦需要对基本速度变量的取值深度局部光滑。数值计算结果表明本文提出的光滑技术可以很好地改善Boussinesq方程模拟浅水波和深水波在斜坡地形上散射问题的能力。     

Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

The observation that the hyperbolic shallow water equations and the Green–Naghdi equations in Lagrangian coordinates have the form of an Euler–Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green–Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.     

变深度浅水域中非定常船波

以Green—Naghdi(G—N)方程为基础，采用波动方程／有限元法计算船舶经过变深度浅水域时非定常波浪特性．把运动船舶对水面的扰动作为移动压强直接加在Green-Naghdi方程里，以描述运动船体和水面的相互作用．以Series60 CB＝0．6船为算例，给出自由面坡高，波浪阻力在船舶经过一个水下凸包时变化规律，并与浅水方程的结果进行了比较．计算结果表明，当船舶经过凸包时，波浪阻力先增加，后减少，并逐渐趋于正常．同时发现，当船速小于临界速度时(Fr＝√gh＜1．0)，G—N方程给出的船后尾波比浅水方程的结果明显，波浪阻力也比浅水方程的结果有所提高，频率散射必须考虑．当船速大于临界速度时(Fr＝√gh＞1．0)，G—N方程的计算结果与浅水方程差别不大，频率散射的影响可以忽略．     

Boundary element calculation of shallow water waves

A boundary element method is proposed for studying periodic shallow water problems. The numerical model is based on the shallow water equation. The key feature of this method is that the boundary integral equations are derived using the weighted residual method and the fundamental solutions for shallow water wave problems are obtained by solving the simultaneous singular equations. The accuracy of this method is studied for the wave reflection problem in a rectangular tank. As a result of this test, it has been shown that the number of element divisions and the distribution of nodes are significant to the accuracy. For numerical examples of external problems, the wave diffraction problems due to single cylindrical, double cylindrical and plate obstructions are analysed and compared with the exact and other numerical solutions. Relatively accurate solutions are obtained.     

Similarities between the quasi‐bubble and the generalized wave continuity equation solutions to the shallow water equations

Two common strategies for solving the shallow water equations in the finite element community are the generalized wave continuity equation (GWCE) reformulation and the quasi‐bubble velocity approximation. The GWCE approach has been widely analysed in the literature. In this work, the quasi‐bubble equations are analysed and comparisons are made between the quasi‐bubble approximation of the primitive form of the shallow water equations and a linear finite element approximation of the GWCE reformulation of the shallow water equations. The discrete condensed quasi‐bubble continuity equation is shown to be identical to a discrete wave equation for a specific GWCE weighting parameter value. The discrete momentum equations are slightly different due to the bubble function. In addition, the dispersion relationships are shown to be almost identical and numerical experiments confirm that the two schemes compute almost identical results. Analysis of the quasi‐bubble formulation suggests a relationship that may guide selection of the optimal GWCE weighting parameter. Copyright © 2004 John Wiley & Sons, Ltd.     

An integrated modelling system for coastal area dynamics

Two-dimensional initial-boundary value problems are considered for the shallow water equations and the equation of advection and dispersion of pollutants. The problems are solved in curvilinear boundary fitted co-ordinates. The transformed equations are integrated on a regular grid by the semi-implicit and implicit finite difference methods. Based on the numerical method, the integrated modelling system Cardinal for coastal area dynamics and pollution processes is developed for application on personal computers. Examples of computations are given.     

Method of relaxed streamline upwinding for hyperbolic conservation laws

In this work a new finite element based Method of Relaxed Streamline Upwinding is proposed to solve hyperbolic conservation laws. Formulation of the proposed scheme is based on relaxation system which replaces hyperbolic conservation laws by semi-linear system with stiff source term also called as relaxation term. The advantage of the semi-linear system is that the nonlinearity in the convection term is pushed towards the source term on right hand side which can be handled with ease. Six symmetric discrete velocity models are introduced in two dimensions which symmetrically spread foot of the characteristics in all four quadrants thereby taking information symmetrically from all directions. Proposed scheme gives exact diffusion vectors which are very simple. Moreover, the formulation is easily extendable from scalar to vector conservation laws. Various test cases are solved for Burgers equation (with convex and non-convex flux functions), Euler equations and shallow water equations in one and two dimensions which demonstrate the robustness and accuracy of the proposed scheme. New test cases are proposed for Burgers equation, Euler and shallow water equations. Exact solution is given for two-dimensional Burgers test case which involves normal discontinuity and series of oblique discontinuities. Error analysis of the proposed scheme shows optimal convergence rate. Moreover, spectral stability analysis gives implicit expression of critical time step.     

浅水方程组合型超紧致差分格式

提出一族组合型超紧致差分格式(CSCD),对CSCD的数值特性作了分析,并同其他中心型差分格式进行比较。从定性角度,得出同阶中心差分格式中,CSCD格式的截断误差系数最小的结论。从定量角度,利用Fou-rier分析方法分析了CSCD格式的分辨率,并同其他中心型差分格式比较,得出CSCD格式有较高的分辨率的结论。把10阶CSCD格式应用于KdV-Burgers方程和浅水方程的数值模拟,给出两个应用算例。数值实验表明CSCD格式不仅有理论上的高精度,而且有良好的稳定性和收敛性。     

Bathymetry reconstruction based on the zero-inertia shallow water approximation

The knowledge of the channel bed topography is paramount in modeling the hydrodynamics of open channel flows. Indeed, flow models based on the Shallow Water Approximation require prior information on the channel bed topography to accurately capture the flow features in natural rivers, estuaries, and flood plains. We present here a numerical technique for reconstructing the channel bed topography from given free surface elevation data for steep open channel flows for which the zero-inertia shallow water approximation holds. In this context, the shallow water equations are modified by neglecting inertia terms while retaining the effects of the bed slope and friction terms. We show in this work that by algebraic manipulation, we can recast the governing equations into a single first-order partial differential equation which describes the inverse problem which consists in finding the bed topography from known free surface elevation data. Interestingly, the analysis shows that the inverse problem does not require the knowledge of the bed roughness. The forward problem is solved using MacCormack’s explicit numerical scheme by considering unsteady modified shallow water equations. However, the inverse problem is solved using the method of characteristics. The results of the inverse and the forward problem are successfully tested against each other on two different test cases.