In this paper, we investigate steady states of shallow water moment equations including bottom topographies. We derive a new hyperbolic shallow water moment model based on linearized moment equations that allows for a simple assessment of the steady states. After proving hyperbolicity of the new model, the steady states are fully identified. A well-balanced scheme is adopted to the specific structure of the new model and allows to preserve the steady states in numerical simulations.
The Riemann problem for the shallow water equations with discontinuous topography
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université de Paris 6（雅克-路易·利翁斯实验室与法国国家科学研究中心，巴黎第六大学）；Department of Mathematics, International University（国际大学数学系）
A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique Université Pierre et Marie Curie (Paris 6)（雅克-路易·利翁斯实验室 & 法国国家科学研究中心 皮埃尔和玛丽·居里大学）；Department of Mathematics, International University（国际大学数学系）
We carry out the group classification of the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The equivalence group of this class is found by the algebraic method. Using algebraic techniques, we construct additional point equivalences between some of the listed cases of Lie-symmetry extensions, which are inequivalent up to transformations from the equivalence group.
The Riemann problem of one dimensional shallow water equations with discontinuous topography has been constructed recently. The elementary waves include shock waves, rarefaction waves, and the stationary wave. The stationary wave appears when the water depth changes, especially when there exists a bottom step. In this paper, we are mainly concerned with the interaction between a stationary wave with either a shock wave or a rarefaction wave. By using the characteristic analysis methods, the evolution of waves is described during the interaction process. The solution in large time scale is also…
We classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the space of zeroth-order conservation laws, which are inequivalent up to transformations from the equivalence group. Hamiltonian structures of systems of shallow water equations are used for relating th…
We present an adaptive well-balanced positivity preserving central-upwind scheme on quadtree grids for shallow water equations. The use of quadtree grids results in a robust, efficient and highly accurate numerical method. The quadtree model is developed based on the well-balanced positivity preserving central-upwind scheme proposed in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5 (2007), pp. 133--160]. The designed scheme is well-balanced in the sense that it is capable of exactly preserving "lake-at-rest" steady states. In order to achieve this as well as to preserve positivity of water…
We investigate the Riemann Problem for a shallow water model with porosity and terrain data. Based on recent results on the local existence, we build the solution in the large settings (the magnitude of the jump in the initial data is not supposed to be ``small enough''). One difficulty for the extended solution arises from the double degeneracy of the hyperbolic system describing the model. Another difficulty is given by the fact that the construction of the solution assumes solving an equation which has no global solution. Finally, we present some cases to illustrate the existence and non-ex…
We develop a new finite volume method using unstructured mesh-vertex grids for coupled systems modeling shallow water flows and solute transport over complex bottom topography. Novel well-balanced positivity preserving discretization techniques are proposed for the water surface elevation and the concentration of the pollutant. For the hydrodynamic system, the proposed scheme preserves the steady state of a lake at rest and the positivity of the water depth. For the scalar transport equation, the proposed method guarantees the positivity and a perfect balance of the scalar concentration. The c…
The present work is devoted to the derivation of a fully well-balanced and positivepreserving numerical scheme for the shallow water equations with Coriolis force. The first main issue consists in preserving all the steady states, including the geostrophic equilibrium. Our strategy relies on a Godunov-type scheme with suitable source term and steady state discretisations. The second challenge lies in improving the order of the scheme while preserving the fully well-balanced property. A modification of the classical methods is required since no conservative reconstruction can preserve all the s…
We consider the two-dimensional shallow water model derived by Levermore and Sammartino (Nonlinearity 14,2001), describing the motion of an incompressible fluid, confined in a shallow basin, with varying bottom topography. We construct the approximate inertial manifolds for the associated dynamical system and estimate its order. Finally, considering the whole domain R^2 and under suitable conditions on the time dependent forcing term, we prove the L^2 asymptotic decay of the weak solutions.
Minimizing computational cost is one of the major challenges in the modelling and numerical analysis of hydrodynamics, and one of the ways to achieve this is by the use of quadtree grids. In this paper, we present an adaptive scheme on quadtree grids for variable density shallow water equations. A scheme for the coupled system is developed based on the well-balanced positivity-preserving central-upwind scheme proposed in [18]. The scheme is capable of exactly preserving "lake-at-rest" steady states. A continuous piecewise bi-linear interpolation of the bottom topography function is used to ach…
We prove in this paper a long time existence result for a modified Boussinesq-Peregrine equation in one dimension, describing the motion of Water Waves in shallow water, in the case of a non flat bottom. We first give a local existence result for the original Boussinesq Peregrine equation as derived by Boussinesq and Peregrine in all dimensions. We then introduce a new model which has formally the same precision as the Boussinesq-Peregrine equation, and give a local existence result in all dimensions. We finally prove a long time existence result in dimension 1 for this new equation, without a…
Overland flow on agricultural fields may have some undesirable effects such as soil erosion, flood and pollutant transport. To better understand this phenomenon and limit its consequences, we developed a code using state-of-the-art numerical methods: FullSWOF (Full Shallow Water equations for Overland Flow), an object oriented code written in C++. It has been made open-source and can be downloaded from http://www.univ-orleans.fr/mapmo/soft/FullSWOF/. The model is based on the classical system of Shallow Water (SW) (or Saint-Venant system). Numerical difficulties come from the numerous dry/wet …
The Green-Naghdi equations are a nonlinear dispersive perturbation of the nonlinear shallow water equations, more precise by one order of approximation. These equations are commonly used for the simulation of coastal flows, and in particular in regions where the water depth vanishes (the shoreline). The local well-posedness of the Green-Naghdi equations (and their justification as an asymptotic model for the water waves equations) has been extensively studied, but always under the assumption that the water depth is bounded from below by a positive constant. The aim of this article is to remove…
We are interested in asymptotic models for the propagation of internal waves at the interface between two shallow layers of immiscible fluid, under the rigid-lid assumption. We review and complete existing works in the literature, in order to offer a unified and comprehensive exposition. Anterior models such as the shallow water and Boussinesq systems, as well as unidirectional models of Camassa-Holm type, are shown to descend from a broad Green-Naghdi model, that we introduce and justify in the sense of consistency. Contrarily to earlier works, our Green-Naghdi model allows a non-flat topogra…
The derivation of shallow water models from Navier-Stokes equations is revisited yielding a class of two-layer shallow water models.An improved velocity profile is proposed, based on the superposition of an ideal fluid and a viscous layer inspired by the Interactive Boundary Layer interaction used in aeronautics. This leads to a new friction law which depends not only on velocity and depth but of the variations of velocity and thickness of boundary layer. The resulting system is an extended shallow water model consisting of three depth-integrated equations: the first two are mass and momentum …
Consideration is given to the influence of an underwater landslide on waves at the surface of a shallow body of fluid. The equations of motion which govern the evolution of the barycenter of the landslide mass include various dissipative effects due to bottom friction, internal energy dissipation, and viscous drag. The surface waves are studied in the Boussinesq scaling, with time-dependent bathymetry. A numerical model for the Boussinesq equations is introduced which is able to handle time-dependent bottom topography, and the equations of motion for the landslide and surface waves are solved …
We study here the propagation of long waves in the presence of vorticity. In the irrotational framework, the Green-Naghdi equations (also called Serre or fully nonlinear Boussinesq equations) are the standard model for the propagation of such waves. These equations couple the surface elevation to the vertically averaged horizontal velocity and are therefore independent of the vertical variable. In the presence of vorticity, the dependence on the vertical variable cannot be removed from the vorticity equation but it was however shown in [?] that the motion of the waves could be described using …
