. Some approximate Godunov schemes to compute shallow-water equations with topography. Computers and Fluids, Elsevier, 2003, 32 (4) AbstractWe study here the computation of shallow-water equations with topography by Finite Volume methods, in a one-dimensional framework (though all methods introduced may be naturally extended in two dimensions). All methods performed are based on a dicretisation of the topography by a piecewise function constant on each cell of the mesh, from an original idea of A.Y. Le Roux et al.. Whereas the Well-Balanced scheme of A.Y. Le Roux is based on the exact resolution of each Riemann problem, we consider here approximate Riemann solvers, namely the VFRoencv schemes. Several single step methods are derived from this formalism, and numerical results are compared to a fractional step method. Some tests cases are presented : convergence to steady states in subcritical and supercritical con gurations, occurence of dry area by a drain over a bump and occurence of vacuum by a double rarefaction wave over a step. Numerical schemes, combined with an appropriate high order extension, provide accurate and convergent approximations.
This paper is devoted to the numerical analysis of the road traffic model proposed by Colombo and Goatin in [CG07]. The model involves a standard conservation law supplemented by a local unilateral constraint on the flux at the point x = 0 (modelling a road light, a toll gate, etc.).We first show that the problem can be interpreted in terms of the theory of conservation laws with discontinuous flux function, as developed by Adimurthi et al.[AMG05] and Bürger et al. [BKT09]. We reformulate accordingly the notion of entropy solution introduced in [CG07], and extend the well-posedness results to the L ∞ framework. Then, starting from a general monotone finite volume scheme for the non-constrained conservation law, we produce a simple scheme for the constrained problem and show its convergence. The proof uses a new notion of entropy process solution. Numerical examples modelling a "green wave" are presented.
Closure laws for interfacial pressure and interfacial velocity are proposed within the frame of two-pressure two-phase flow models. These enable to ensure positivity of void fractions, mass fractions and internal energies when investigating field by field waves in the Riemann problem. Lois de fermeture pour un modèleà deux pressions d'écoulement diphasique Résumé. On propose des lois de fermeture de vitesse et de pression d'interface pour un modèle d'écoulement diphasiqueà deux pressions. Celles-ci assurent champ par champ de respecter la positivité des fractions volumiques, des variables densité eténergie interne si on examine le problème de Riemann.
In this paper, we present an original derivation process of a nonhydrostatic shallow water-type model which aims at approximating the incompressible Euler and Navier-Stokes systems with free surface. The closure relations are obtained by a minimal energy constraint instead of an asymptotic expansion. The model slightly differs from the well-known Green-Naghdi model and is confronted with stationary and analytical solutions of the Euler system corresponding to rotational flows. At the end of the paper, we give time-dependent analytical solutions for the Euler system that are also analytical solutions for the proposed model but that are not solutions of the Green-Naghdi model. We also give and compare analytical solutions of the two non-hydrostatic shallow water models.2010 Mathematics Subject Classification. 35L60, 35Q30, 76B15, 76D05. simulations of a large class of geophysical phenomena such as rivers, lava flows, ice sheets, coastal domains, oceans or even run-off or avalanches when being modified with adapted source terms [6,7,30]. But the Saint-Venant system is built on the hydrostatic assumption consisting in neglecting the vertical acceleration of the fluid. This assumption is valid for a large class of geophysical flows but is restrictive in various situations where the dispersive effects -such as those occuring in wave propagation -cannot be neglected. As an example, neglecting the vertical acceleration in granular flows or landslides leads to significantly overestimate the initial flow velocity [31, 28], with strong implication for hazard assessment.The modeling of the non-hydrostatic effects for shallow water flows does not raise insuperable difficulties [20,12,4,36,37, 10] but the analysis [1,26] of the resulting models and their discretization become tough. The assumption of potential flows is often used to derive dispersive models and an extensive literature exists concerning these models. The most important contributions have been proposed by Lannes and co-authors [5,13,22,1,2], see also [16].The non-hydrostatic model presented in this paper is not based on the irrotational assumption, on the other hand it is not derived using an asymptotic expansion of the incompressible Navier-Stokes or Euler based on the classical shallow water assumptions. Even if such an asymptotic expansion approach is natural, it leads to difficulties for the approximation of the non-hydrostatic pressure terms.To overcome these problems, we propose a strategy for the model derivation that is widely used in the kinetic framework to obtain kinetic descriptions e.g. of conservations laws [25,38]. The required closure relations to obtain a depthaveraged model approximating the Euler or Navier-Stokes system satisfy an energybased optimality criterion. As a consequence, the proposed model slightly differs from existing models especially the well-known Green-Naghdi model [20,26]. It consists in a set of first order partial differential equations and compared to the Green-Naghdi model, the contribution of the non-hydrostatic pressure...
SUMMARYWe consider numerical solutions of the two-dimensional non-linear shallow water equations with a bed slope source term. These equations are well-suited for the study of many geophysical phenomena, including coastal engineering where wetting and drying processes are commonly observed. To accurately describe the evolution of moving shorelines over strongly varying topography, we first investigate two well-balanced methods of Godunov-type, relying on the resolution of non-homogeneous Riemann problems. But even if these schemes were previously proved to be efficient in many simulations involving occurrences of dry zones, they fail to compute accurately moving shorelines. From this, we investigate a new model, called SURF WB, especially designed for the simulation of wave transformations over strongly varying topography. This model relies on a recent reconstruction method for the treatment of the bed-slope source term and is able to handle strong variations of topography and to preserve the steady states at rest. In addition, the use of the recent VFRoe-ncv Riemann solver leads to a robust treatment of wetting and drying phenomena. An adapted 'second order' reconstruction generates accurate bore-capturing abilities. This scheme is validated against several analytical solutions, involving varying topography, time dependent moving shorelines and convergences toward steady states. This model should have an impact in the prediction of 2D moving shorelines over strongly irregular topography.
SUMMARYThis paper deals with the resolution by ÿnite volume methods of Euler equations in one space dimension, with real gas state laws (namely, perfect gas EOS, Tammann EOS and Van Der Waals EOS). All tests are of unsteady shock tube type, in order to examine a wide class of solutions, involving Sod shock tube, stationary shock wave, simple contact discontinuity, occurrence of vacuum by double rarefaction wave, propagation of a one-rarefaction wave over 'vacuum', ... Most of the methods computed herein are approximate Godunov solvers: VFRoe, VFFC, VFRoe ncv ( ; u; p) and PVRS. The energy relaxation method with VFRoe ncv ( ; u; p) and Rusanov scheme have been investigated too. Qualitative results are presented or commented for all test cases and numerical rates of convergence on some test cases have been measured for ÿrst-and second-order (Runge-Kutta 2 with MUSCL reconstruction) approximations. Note that rates are measured on solutions involving discontinuities, in order to estimate the loss of accuracy due to these discontinuities.
We study here a model of conservative nonlinear conservation law with a flux function with discontinuous coefficients, namely the equation ut + (k(x)u(1 - u))x = 0. It is a particular entropy condition on the line of discontinuity of the coefficient k which ensures the uniqueness of the entropy solution. This condition is discussed and justified. On the other hand, we perform a numerical analysis of the problem. Two finite volume schemes, the Godunov scheme and the VFRoe-ncv scheme, are proposed to simulate the conservation law. They are compared with two finite volume methods classically used in an industrial context. Several tests confirm the good behavior of both new schemes, especially through the discontinuity of permeability k (whereas a loss of accuracy may be detected when industrial methods are performed). Moreover, a modified MUSCL method which accounts for stationary states is introduced.
We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions. The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds. In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing the robustness and stability of the method in the limits of small phase fractions. The scheme is proved to satisfy a discrete entropy inequality and to preserve positive values of the statistical fractions and densities. The numerical simulations show a much higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry. Finally, two test-cases assess the good behavior of the scheme when approximating vanishing phase solutions.
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