一个求解浅水波方程的时域分段展开算法

发展了一种时域分段展开自适应方法求解一维非线性浅水波方程。通过时域分段展开,将一个非线性的时空耦合初边值问题转化为一系列的线性空间边值问题,并采用有限元方法递推求解;通过展开阶数的递进,实现了分段时域的自适应计算,当不同步长时可保持稳定的计算精度。研究结果表明,当步长较大而Heun’s法、四阶Runge-Kutta法不能得到合理结果时,本文算法仍能保证足够的计算精度。     

求解浅水波方程的半离散中心迎风方法

将Jin's的界面方法应用到求解双曲守恒型方程的半离散中心迎风方法中,给出了一种新的求解浅水波方程的半离散中心迎风差分方法。对于源项,不是采用传统的单元均值而是采用单元界面处的值来近似,使所得格式对稳定态的求解是均衡的。且已证明所给的二阶精度的求解格式保持水深的非负性,这一特性使其能够较好的处理干河床问题。使用该方法产生的数值粘性(与O(Δ2r-1)同阶)要比交错的中心格式小(与O(Δx2r/Δt)同阶),而且由于数值粘性与时间步长无关,从而时间步长可根据稳定性需要尽可能的小,因此适用于稳定态的求解。     

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。     

Whitham—Broer—Kaup浅水波方程新的多孤子解

Whitham-Broer-Kaup（简记WBK）方程具有重要的意义,至今人们只给出了它的单孤子解,本利用齐次平衡法并借助数学给出它新的多孤子解,并作为特例得到一类变式Boussinesq方程的多孤子解。     

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。     

基于非结构网格求解二维浅水方程的高精度有限体积方法

采用HLL格式,在三角形非结构网格下采用有限体积离散,建立了求解二维浅水方程的高精度的数值模型.本文采用多维重构和多维限制器的方法来获得高精度的空间格式以及防止非物理振荡的产生,时间离散采用三阶Runge-Kutta法以获得高阶的时间精度.基于三角形网格,底坡源项采用简单的斜底模型离散,为保证计算格式的和谐性,对经典的HLL格式计算的数值通量中的静水压力项进行了修正.算例证明本文提出的方法的和谐性并具有高精度的间断捕捉能力和稳定性.     

浅水孤立波在三维浮体上的绕射

浅水域中非线性水波运动的控制方程通常是经过深度平均的Boussinesq方程。然而,这一方程在浮体近旁或水下障碍物附近不再适用,在这些区域,流动在水深方向的变化不容忽略,本文应用匹配渐近展开法和边缘层(edge layer)思想,建立了浅水弱非线性波与三维浮体相互作用的数学模型,作为算例,求解了浅水孤立波在垂直圆柱形浮体上的绕射.本方法可以推广到波在一般浮体上绕射的情况。     

基于神经网络的差分方程快速求解方法

近年来, 人工神经网络(artificial?neural?networks, ANN), 尤其是深度神经网络(deep?neural?networks, DNN)由于其在异构平台上的高计算效率与对高维复杂系统的拟合能力而成为一种在数值计算领域具有广阔前景的新方法. 在偏微分方程数值求解中, 大规模线性方程组的求解通常是耗时最长的步骤之一, 因此, 采用神经网络方法求解线性方程组成为了一种值得期待的新思路. 但是, 深度神经网络的直接预测仍在数值精度方面仍有明显的不足, 成为其在数值计算领域广泛应用的瓶颈之一. 为打破这一限制, 本文提出了一种结合残差网络结构与校正迭代方法的求解算法. 其中, 残差网络结构解决了深度网络模型的网络退化与梯度消失等问题, 将网络的损失降低至经典网络模型的1/5000; 修正迭代的方法采用同一网络模型对预测解的反复校正, 将预测解的残差下降至迭代前的10?5倍. 为验证该方法的有效性与通用性, 本文将该方法与有限差分法结合, 对热传导方程与伯格方程进行了求解. 数值结果表明, 本文所提出的算法对于规模大于1000的方程组具有10倍以上的加速效果, 且数值误差低于二阶差分格式的离散误差.      

非圆截面杆中非线性扭转波方程的解析求解

研究了非圆截面杆中非线性扭转波动方程的精确求解问题.利用直接积分与微分变换相结合的方法,得到了该方程的隐式通解.通过对积分常数和方程系数的不同情形的讨论,给出了该方程三角函数、双曲函数、椭圆函数、指数函数以及它们的组合形式的解,分别对应于非线性扭转波的孤立波、周期波以及冲击波等多种传播形式.     

一维孔隙率浅水方程及其数值离散

通过孔隙率方法来描述挡水物对过水能力的影响建立了一维孔隙率浅水方程. 采用有限体积方法和Roe格式的近似Riemann解建立了孔隙率浅水方程的离散模式. 对底坡和孔隙率源项采用特性方向分解的方法进行处理,使模型精确满足C(Conservative)特性,增加了模型的稳定性. 通过算例模拟证明了模型可以对河道中的挡水物作用进行模拟,且计算结果表明模型具有和谐、稳定、分辨率高等优点.      

An adaptive artificial viscosity for the displacement shallow water wave equation

The numerical oscillation problem is a difficulty for the simulation of rapidly varying shallow water surfaces which are often caused by the unsmooth uneven bottom,the moving wet-dry interface, and so on. In this paper, an adaptive artificial viscosity(AAV) is proposed and combined with the displacement shallow water wave equation(DSWWE) to establish an effective model which can accurately predict the evolution of multiple shocks effected by the uneven bottom and the wet-dry interface. The effec...     

On the approximation of local efflux/influx bed discharge in the shallow water equations based on a wave propagation algorithm

In cities, flood waves may propagate over street surfaces below which lie complicated pipe networks used for storm drainage and sewage. The flood and pipe flows can interact at connections between the underground pipes and the street surface. The present paper examines this interaction, using the shallow water equations to model the flood wave hydrodynamics. Sources and sinks in the mass conservation equation are used to model the pipe inflow and outflow conditions at bed connections. We consider the problem reduced to one dimension. The shallow water equations are solved using a Godunov‐type wave propagation scheme. Wave speeds are modified in the wave propagation algorithm to enable flows to be simulated over nearly dry beds and dry states. First, the model is used to simulate vertical flows through finite gaps in the bed. Next, the interaction of the vertical flows with a dam break flow is considered for both dry and wet beds. An efflux number, En, is defined based on the vertical efflux velocity and the gap length. Comparisons are made with numerical predictions from STAR‐CD, a commercial Navier–Stokes solver that models the free‐surface motions, and a parameter study is undertaken to investigate the effect of the one‐dimensional approximation of the present model, for a range of non‐dimensional efflux numbers. It is found that the shallow flow model gives sensible predictions at all time provided En<0.5, and for long durations for En>0.5. Dam break flow over an underground connecting pipe is also considered. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

A simplified adaptive Cartesian grid system for solving the 2D shallow water equations

This paper presents a new simplified grid system that provides local refinement and dynamic adaptation for solving the 2D shallow water equations (SWEs). Local refinement is realized by simply specifying different subdivision levels to the cells on a background uniform coarse grid that covers the computational domain. On such a non‐uniform grid, the structured property of a regular Cartesian mesh is maintained and neighbor information is determined by simple algebraic relationships, i.e. data structure becomes unnecessary. Dynamic grid adaptation is achieved by changing the subdivision level of a background cell. Therefore, grid generation and adaptation is greatly simplified and straightforward to implement. The new adaptive grid‐based SWE solver is tested by applying it to simulate three idealized test cases and promising results are obtained. The new grid system offers a simplified alternative to the existing approaches for providing adaptive mesh refinement in computational fluid dynamics. Copyright © 2011 John Wiley & Sons, Ltd.     

A 2D implicit time‐marching algorithm for shallow water models based on the generalized wave continuity equation

This paper builds upon earlier work that developed and evaluated a 1D predictor–corrector time‐marching algorithm for wave equation models and extends it to 2D. Typically, the generalized wave continuity equation (GWCE) utilizes a three time‐level semi‐implicit scheme centred at k, and the momentum equation uses a two time‐level scheme centred at k+12. It has been shown that in highly non‐linear applications, the algorithm becomes unstable at even moderate Courant numbers. This work implements and analyses an implicit treatment of the non‐linear terms through the use of an iterative time‐marching algorithm in the two‐dimensional framework. Stability results show at least an eight‐fold increase in the maximum time step, depending on the domain. Studies also examined the sensitivity of the G parameter (a numerical weighting parameter in the GWCE) with results showing the greatest increase in stability occurs when 1?G/τ_max?10, a range that coincides with the recommended range to minimize errors. Convergence studies indicate an increase in temporal accuracy from first order to second order, while overall error is less than the original algorithm, even at higher time steps. Finally, a parallel implementation of the new algorithm shows that it scales well. Copyright © 2004 John Wiley & Sons, Ltd.     

Sparse element for shallow water wave equations on triangle meshes

Within the mixed FEM, the mini‐element that uses a bubble shape function for the solution of the shallow water wave equations on triangle meshes is simplified to a sparse element formulation. The new formulation has linear shape functions for water levels and constant shape functions for velocities inside each element. The suppression of decoupled spurious solutions is excellent with the new scheme. The linear dispersion relation of the new element has similar advantages as that of the wave equation scheme (generalised wave continuity scheme) proposed by Lynch and Gray. It is shown that the relation is monotonic over all wave numbers. In this paper, the time stepping scheme is included in the dispersion analysis. In case of a combined space–time staggering, the dispersion relation can be improved for the shortest waves. The sparse element is applied in the flow model Bubble that conserves mass exactly. At the same time, because of the limited number of degrees of freedom, the computational efficiency is high. The scheme is not restricted to orthogonal triangular meshes. Three test cases demonstrate the very good accuracy of the proposed scheme. The examples are the classical quarter annulus test case for the linearised shallow water equations, the hydraulic jump and the tide in the Elbe river mouth. Copyright © 2013 John Wiley & Sons, Ltd.     

A meshless numerical wave tank for simulation of nonlinear irregular waves in shallow water

Time domain simulation of the interaction between offshore structures and irregular waves in shallow water becomes a focus due to significant increase of liquefied natural gas (LNG) terminals. To obtain the time series of irregular waves in shallow water, a numerical wave tank is developed by using the meshless method for simulation of 2D nonlinear irregular waves propagating from deep water to shallow water. Using the fundamental solution of Laplace equation as the radial basis function (RBF) and locating the source points outside the computational domain, the problem of water wave propagation is solved by collocation of boundary points. In order to improve the computation stability, both the incident wave elevation and velocity potential are applied to the wave generation. A sponge damping layer combined with the Sommerfeld radiation condition is used on the radiation boundary. The present model is applied to simulate the propagation of regular and irregular waves. The numerical results are validated by analytical solutions and experimental data and good agreements are observed. Copyright © 2008 John Wiley & Sons, Ltd.