A new shallow water model with polynomial dependence on depth

In this paper, we study two‐dimensional Euler equations in a domain with small depth. With this aim, we introduce a small non‐dimensional parameter ε related to the depth and we use asymptotic analysis to study what happens when ε becomes small. We obtain a model for ε small that, after coming back to the original domain, gives us a shallow water model that considers the possibility of a non‐constant bottom, and the horizontal velocity has a dependence on z introduced by the vorticity when it is not zero. This represents an interesting novelty with respect to shallow water models found in the literature. We stand out that we do not need to make a priori assumptions about velocity or pressure behaviour to obtain the model. The new model is able to approximate the solutions to Euler equations with dependence on z (reobtaining the same velocities profile), whereas the classic model just obtains the average velocity. Copyright © 2007 John Wiley & Sons, Ltd.     

On a numerical flux for the shallow water equations

The finite volume scheme of Vijayasundaram and Osher-Solomon type for shallow water equations are proposed. The numerical results with discontinuous initial condition and the comparison with Lax-Friedrichs numerical flux are presented for homogeneous case. The extension of the method for the inhomogeneous case is described.     

On the local wellposedness of 3-D water wave problem with vorticity

In this article, we first present an equivalent formulation of the free boundary problem to 3-D incompressible Euler equations, then we announce our local wellposedness result concerning the free boundary problem in Sobolev space provided that there is no self-intersection point on the initial surface and under the stability assumption that (δ)p/(δ)n (ξ)|t=0 ≤ -2c0 ＜ 0 with ξ being restricted to the initial surface.     

A new shallow water model,by asymptotic analysis,with linear dependence on depth

In this paper, we study Euler equations in a domain with small depth. With this aim, we introduce a small a-dimensional parameter ε related to the depth and we use asymptotic analysis to study what happens when ε becomes small. In this way we obtain a shallow water model that considers the possibility of a non-constant bottom and the horizontal velocity components depend on z if the vorticity is not zero. The new model is able to calculate exactly the solutions to Euler equations that are linear in z, whereas the classic model just obtains the average velocities. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the dynamics of Navier-Stokes equations for a shallow water model

In this paper, we study a free boundary problem of one-dimensional compressible Navier-Stokes equations with a density-dependent viscosity, which include, in particular, a shallow water model. Under some suitable assumptions on the initial data, we obtain the global existence, uniqueness and the large time behavior of weak solutions. In particular, it is shown that a stationary wave pattern connecting a gas to the vacuum continuously is asymptotically stable for small initial general perturbations.     

Self-similar point vortices and confinement of vorticity

This papers deals with the large time behavior of solutions of the incompressible Euler equations in dimension 2. We consider a self-similar configuration of point vortices which grows like the square root of the time. We study the confinement properties of a blob of vorticity initially located around the first point vortex and moving in the velocity field produced by itself and by the other point vortices. We find a su?cient condition on the point vortices such that the vorticity stays confined around the first point vortex at a rate better than the square root of the time. The relevance to the large time behavior of the Euler equations is discussed.     

On the shallow water equations with periodicity in one direction

The linearized 2D shallow water equations, supplemented with suitable boundary conditions in one direction and periodicity in the other direction are considered. The well-posedness of this mixed boundary value problem is proven, using the linear semi-group theory. The well-posedness of the totally periodic problem is also proven.     

Simulation of the spread of a viscous fluid using a bidimensional shallow water model

In this paper we propose a numerical method to solve the Cauchy problem based on the viscous shallow water equations in an horizontally moving domain. More precisely, we are interested in a flooding and drying model, used to modelize the overflow of a river or the intrusion of a tsunami on ground. We use a nonconservative form of the two-dimensional shallow water equations, in eight velocity formulation and we build a numerical approximation, based on the Arbitrary lagrangian eulerian formulation, in order to compute the solution in the moving domain.     

Method for solving hyperbolic systems with initial data on non-characteristic manifolds with applications to the shallow water wave equations

We are concerned with hyperbolic systems of order-one linear PDEs originated on non-characteristic manifolds. We put forward a simple but effective method of transforming such initial conditions to standard initial conditions (i.e. when the solution is specified at an initial moment of time). We then show how our method applies in fluid mechanics. More specifically, we present a complete solution to the problem of long waves run-up in inclined bays of arbitrary shape with nonzero initial velocity.     

A simple finite volume method for the shallow water equations

We present a new finite volume method for the numerical solution of shallow water equations for either flat or non-flat topography. The method is simple, accurate and avoids the solution of Riemann problems during the time integration process. The proposed approach consists of a predictor stage and a corrector stage. The predictor stage uses the method of characteristics to reconstruct the numerical fluxes, whereas the corrector stage recovers the conservation equations. The proposed finite volume method is well balanced, conservative, non-oscillatory and suitable for shallow water equations for which Riemann problems are difficult to solve. The proposed finite volume method is verified against several benchmark tests and shows good agreement with analytical solutions.     

Exponential growth of the vorticity gradient for the Euler equation on the torus

We prove that there are solutions to the Euler equation on the torus with C^1,α$C^{1,\alpha }$ vorticity and smooth except at one point such that the vorticity gradient grows in L^∞$L^{\infty }$ at least exponentially as t→∞$t\to \infty$. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev and Šverák [6].     

Well-balanced unstaggered central schemes for one and two-dimensional shallow water equation systems

We propose a new well-balanced unstaggered central finite volume scheme for hyperbolic balance laws with geometrical source terms. In particular we construct a new one and two-dimensional finite volume method for the numerical solution of shallow water equations on flat/variable bottom topographies. The proposed scheme evolves a non-oscillatory numerical solution on a single grid, avoids the time consuming process of solving Riemann problems arising at the cell interfaces, and is second-order accurate both in space and time. Furthermore, the numerical scheme follows a well-balanced discretization that first discretizes the geometrical source term according to the discretization of the flux terms, and then mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The resulting scheme exactly satisfies the C-property at the discrete level. The proposed scheme is then applied and classical one and two-dimensional shallow water equation problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method.     

An existence result for a free boundary shallow water model using a Lagrangian scheme

We propose a free boundary shallow water model for which we give an existence theorem. The proof uses an original Lagrangian discrete scheme in order to build a sequence of approximate solutions. The properties of this scheme allow to treat the difficulties linked to the boundary motion. These approximate solutions verify some compactness results which allow us to pass to the limit in the discrete problem.     

Meshless method for shallow water equations with free surface flow

Solving problems with free surface often encounters numerical difficulties related to excessive mesh distortion as is the case of dambreak or breaking waves. In this paper the Natural element method (NEM) is used to simulate a 2D shallow water flows in the presence of theses strong gradients. This particle-based method used a fully Lagrangian formulation based on the notion of natural neighbors. In the present study we consider the full non-linear set of Shallow Water Equations, with a transient flow under the Coriolis effect. For the numerical treatment of the nonlinear terms we used a Lagrangian technique based on the method of characteristics. This will allow avoiding divergence of Newton-Raphson scheme, when dealing with the convective terms. We also define a thin area close to the boundaries and a computational domain dedicated for nodal enrichment at each time step. Two numerical test cases were performed to verify the well-founded hopes for the future of this method in real applications.     

Weak local residuals as smoothness indicators for the shallow water equations

The system of shallow water equations admits infinitely many conservation laws. This paper investigates weak local residuals as smoothness indicators of numerical solutions to the shallow water equations. To get a weak formulation, a test function and integration are introduced into the shallow water equations. We use a finite volume method to solve the shallow water equations numerically. Based on our numerical simulations, the weak local residual of a simple conservation law with a simple test function is identified to be the best as a smoothness indicator.     

High-order discontinuous element-based schemes for the inviscid shallow water equations: Spectral multidomain penalty and discontinuous Galerkin methods

Two commonly used types of high-order-accuracy element-based schemes, collocation-based spectral multidomain penalty methods (SMPM) and nodal discontinuous Galerkin methods (DGM), are compared in the framework of the inviscid shallow water equations. Differences and similarities in formulation are identified, with the primary difference being the dissipative term in the Rusanov form of the numerical flux for the DGM that provides additional numerical stability; however, it should be emphasized that to arrive at this equivalence between SMPM and DGM requires making specific choices in the construction of both methods; these choices are addressed. In general, both methods offer a multitude of choices in the penalty terms used to introduce boundary conditions and stabilize the numerical solution. The resulting specialized class of SMPM and DGM are then applied to a suite of six commonly considered geophysical flow test cases, three linear and three non-linear; we also include results for a classical continuous Galerkin (i.e., spectral element) method for comparison. Both the analysis and numerical experiments show that the SMPM and DGM are essentially identical; both methods can be shown to be equivalent for very special choices of quadrature rules and Riemann solvers in the DGM along with special choices in the type of penalty term in the SMPM. Although we only focus our studies on the inviscid shallow water equations the results presented should be applicable to other systems of nonlinear hyperbolic equations (such as the compressible Euler equations) and extendable to the compressible and incompressible Navier-Stokes equations, where viscous terms are included.     

A SIRS epidemic model with infection-age dependence

Based on J. Mena-Lorca and H.W. Hethcote's epidemic model, a SIRS epidemic model with infection-age-dependent infectivity and general nonlinear contact rate is formulated. Under general conditions, the unique existence of its global positive solutions is obtained. Moreover, under more general assumptions than the existing, the existence and asymptotical stability of its equilibria are discussed. In the end, the condition on the stability of endemic equilibrium is verified by a special model.     

Central unstaggered finite volume schemes for hyperbolic systems: Applications to unsteady shallow water equations

A class of central unstaggered finite volume methods for approximating solutions of hyperbolic systems of conservation laws is developed in this paper. The proposed method is an extension of the central, non-oscillatory, finite volume method of Nessyahu and Tadmor (NT). In contrast with the original NT scheme, the method we develop evolves the numerical solution on a single grid; however ghost cells are implicitly used to avoid the resolution of the Riemann problems arising at the cell interfaces. We apply our method and solve classical one and two-dimensional unsteady shallow water problems. Our numerical results compare very well with those obtained using the original NT method, and are in good agreement with corresponding results appearing in the recent literature, thus confirming the efficiency and the potential of the proposed method.     

Existence and regularity of the global attractor for a weakly damped forced shallow water equation in H

In this paper we prove the existence of the global attractor for a weakly damped forced shallow water equation in . Moreover, we show that the global attractor is actually compact in . The main method is the combination of the energy equation, a suitable splitting of the solutions, and bilinear estimates in Bourgain spaces.