(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.     

A ‘well‐balanced’ finite volume scheme for blood flow simulation

We are interested in simulating blood flow in arteries with a one‐dimensional model. Thanks to recent developments in the analysis of hyperbolic system of conservation laws (in the Saint‐Venant shallow water equations context) we will perform a simple finite volume scheme. We focus on conservation properties of this scheme, which were not previously considered. To emphasize the necessity of this scheme, we present how a too simple numerical scheme may induce spurious flows when the basic static shape of the radius changes. On the contrary, the proposed scheme is ‘well‐balanced’: it preserves equilibria of Q = 0. Then examples of analytical or linearized solutions with and without viscous damping are presented to validate the calculations. The influence of abrupt change of basic radius is emphasized in the case of an aneurism. Copyright © 2012 John Wiley & Sons, Ltd.     

Scattering of an incident acoustic wave by an infinite elastic cylinder in a shallow sea

The scattering of sound waves by an isotropic elastic circular cylinder of infinite extent in a water of finite depth is investigated taking into account the shear waves that can exist in addition to compressional waves in scatterers of solid material. The axis of the cylinder is parallel to the water level. The reflected and transmitted energies are plotted for the various values of the radius of the cylinder, and the farfield scattered pressure is obtained for various depth and presented in a graphical form. Copyright © 2008 John Wiley & Sons, Ltd.     

Modulational instability in a full‐dispersion shallow water model

We propose a shallow water model that combines the dispersion relation of water waves and Boussinesq equations, and that extends the Whitham equation to permit bidirectional propagation. We show that its sufficiently small and periodic traveling wave is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value, like the Benjamin‐Feir instability of a Stokes wave. We verify that the associated linear operator possesses infinitely many collisions of purely imaginary eigenvalues, but they do not contribute to instability to the leading order in the amplitude parameter. We discuss the effects of surface tension. The results agree with those from a formal asymptotic expansion and a numerical computation for the physical problem.     

Global well-posedness for the viscous shallow water system with Korteweg type

In this paper, we study the Cauchy problem for a viscous shallow water system with Korteweg type in Sobolev spaces. We first establish the local well-posedness of the solution by using the Friedrich method and compactness arguments. Then, we prove the global existence of the solution to the system for the small initial data.     

Fast Wave Averaging for the Equatorial Shallow Water Equations

The equatorial shallow water equations in a suitable limit are shown to reduce to zonal jets as the Froude number tends to zero. This is a theorem of a singular limit with a fast variable coefficient due to the vanishing of the Coriolis force at the equator. Although it is not possible to get uniform estimates in classical Sobolev spaces (other than L²) by differentiating the system, a new method exploiting the particular structure of the fast coefficient leads to uniform estimates in slightly different functional spaces. The computation of resonances shows that fast waves may interact with a strong external forcing, introduced to mimic the effects of moisture, to create zonal jets.