High‐order boundary conditions for linearized shallow water equations with stratification,dispersion and advection

The two‐dimensional linearized shallow water equations are considered in unbounded domains with density stratification. Wave dispersion and advection effects are also taken into account. The infinite domain is truncated via a rectangular artificial boundary ??, and a high‐order open boundary condition (OBC) is imposed on ??. Then the problem is solved numerically in the finite domain bounded by ??. A recently developed boundary scheme is employed, which is based on a reformulation of the sequence of OBCs originally proposed by Higdon. The OBCs can easily be used up to any desired order. They are incorporated here in a finite difference scheme. Numerical examples are used to demonstrate the performance and advantages of the computational method, with an emphasis is on the effect of stratification. Published in 2004 by John Wiley & Sons, Ltd.     

变系数浅水波方程的精确解

均衡作用法给出了一种求非线性发展方程孤波解的有效方法.利用该方法,运用计算机符号计算,求出了变系数的一般浅水波方程的孤子解.     

SWAMP: A two‐dimensional hydrodynamic and quality modeling platform for shallow waters

Water quality two‐dimensional models are often partitioned into separate modules with separate hydraulic and biological units. In most cases this approach results in poor flexibility whenever the biological dynamics has to be adapted to a specific situation. Conversely, an integrated approach is pursued in this article, producing a two‐dimensional hydraulic‐water quality model, named Shallow Water Analysis and Modeling Program (SWAMP) designed for shallow water bodies. The major objective of the work is to create a comprehensive two‐dimensional water quality assessment tool, based on an open framework and combining easy programming of additional procedures with a user‐friendly interface. The model is based on the numerical solution of the partial differential equations describing advection‐diffusion and biological processes on a two‐dimensional rectangular finite elements mesh. The hydraulics and advection‐diffusion modules model were validated both with experimental tracer data collected at a constructed wetland site and a comparison with a commercial hydrodynamic software, showing good agreement in both cases. Moreover, the model was tested in critical conditions for mass conservation, such as time‐varying wet boundary, showing a considerable numerical robustness. In the last part of the article water quality simulations are presented, though validation data are not yet available. Nevertheless, the observed model response demonstrates general consistency with expected results and the advantages of integrating the hydraulic and quality modules. The interactive graphical user interface (GUI) is also shown to represent a simple and effective connective tool to the integrated package. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 663–687, 2002; DOI 10.1002/num.10014     

A stabilized SPH method for inviscid shallow water flows

In this paper, the smoothed particle hydrodynamics (SPH) method is applied to the solution of shallow water equations. A brief review of the method in its standard form is first described then a variational formulation using SPH interpolation is discussed. A new technique based on the Riemann solver is introduced to improve the stability of the method. This technique leads to better results. The treatment of solid boundary conditions is discussed but remains an open problem for general geometries. The dam‐break problem with a flat bed is used as a benchmark test. Copyright © 2004 John Wiley & Sons, Ltd.     

An upstream flux‐splitting finite‐volume scheme for 2D shallow water equations

An upstream flux‐splitting finite‐volume (UFF) scheme is proposed for the solutions of the 2D shallow water equations. In the framework of the finite‐volume method, the artificially upstream flux vector splitting method is employed to establish the numerical flux function for the local Riemann problem. Based on this algorithm, an UFF scheme without Jacobian matrix operation is developed. The proposed scheme satisfying entropy condition is extended to be second‐order‐accurate using the MUSCL approach. The proposed UFF scheme and its second‐order extension are verified through the simulations of four shallow water problems, including the 1D idealized dam breaking, the oblique hydraulic jump, the circular dam breaking, and the dam‐break experiment with 45° bend channel. Meanwhile, the numerical performance of the UFF scheme is compared with those of three well‐known upwind schemes, namely the Osher, Roe, and HLL schemes. It is demonstrated that the proposed scheme performs remarkably well for shallow water flows. The simulated results also show that the UFF scheme has superior overall numerical performances among the schemes tested. Copyright © 2005 John Wiley & Sons, Ltd.     

(2+1)维浅水波方程的新精确解

对(2+1)维浅水波方程的现有解进行了推广.应用CK方法对方程进行求解,得到方程的Backlund变换公式,将已知解代入公式,求得一些新的精确解,从而推广了浅水渡方程的解.     

Central finite volume methods for ideal and shallow water magnetohydrodynamics

We propose two‐dimensional central finite volume methods based on our multidimensional extensions of Nessyahu and Tadmor's one‐dimensional non‐oscillatory central scheme and a constrained transport‐type method to solve ideal magnetohydrodynamic problems (MHD) and shallow water magnetohydrodynamic problems (SMHD). The main numerical scheme is second‐order accurate both in space and time and uses an original Cartesian grid coupled to a Cartesian‐ or diamond‐staggered dual grid to by‐pass the resolution of the Riemann problems at the cell interfaces. To treat the non‐vanishing magnetic field/flux divergence we have constructed an adaptation of Evans and Hawley's constrained transport method specifically designed for central schemes. Our numerical results show the efficiency and the potential of the scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

1‐D numerical modelling of shallow flows with variable horizontal density

A1‐D numerical model is presented for vertically homogeneous shallow flows with variable horizontal density. The governing equations represent depth‐averaged mass and momentum conservation of a liquid–species mixture, and mass conservation of the species in the horizontal direction. Here, the term ‘species’ refers to material transported with the liquid flow. For example, when the species is taken to be suspended sediment, the model provides an idealized simulation of hyper‐concentrated sediment‐laden flows. The volumetric species concentration acts as an active scalar, allowing the species dynamics to modify the flow structure. A Godunov‐type finite volume scheme is implemented to solve the conservation laws written in a deviatoric, hyperbolic form. The model is verified for variable‐density flows, where analytical steady‐state solutions are derived. The agreement between the numerical predictions and benchmark test solutions illustrates the ability of the model to capture rapidly varying flow features over uniform and non‐uniform bed topography. A parameter study examines the effects of varying the initial density and depth in different regions. Copyright © 2009 John Wiley & Sons, Ltd.     

Unsteady behavior of an elastic articulated beam floating on shallow water

The unsteady behavior of an elastic beam composed of hinged homogeneous sections, which freely floats on the surface of an ideal incompressible fluid, is studied within the framework of the linear shallow water theory. The unsteady behavior of the beam is due to incidence of a localized surface wave or initial deformation. Beam deflection is sought in the form of an expansion with respect to eigenfunctions of oscillations in vacuum with time-dependent amplitudes. The problem is reduced to solving an infinite system of ordinary differential equations for unknown amplitudes. The beam behavior with different actions of the medium and hinge positions is studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 54–65, July–August, 2009.