Finite-Volume Solvers for a Multilayer Saint-Venant System

Finite-Volume Solvers for a Multilayer Saint-Venant System We consider the numerical investigation of two hyperbolic shallow water models. We focus on the treatment of the hyperbolic part. We first recall some efficient finite volume solvers for the classical Saint-Venant system. Then we study their extensions to a new multilayer Saint-Venant system. Finally, we use a kinetic solver to perform some numerical tests which prove that the 2D multilayer Saint-Venant system is a relevant alternative to 3D hydrostatic Navier-Stokes equations.

[1]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[2]  Carlos Parés,et al.  A Q-SCHEME FOR A CLASS OF SYSTEMS OF COUPLED CONSERVATION LAWS WITH SOURCE TERM. APPLICATION TO A TWO-LAYER 1-D SHALLOW WATER SYSTEM , 2001 .

[3]  P. Souganidis,et al.  Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates , 1998 .

[4]  Vladimir Zeitlin,et al.  Frontal geostrophic adjustment, slow manifold and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 1. Theory , 2003, Journal of Fluid Mechanics.

[5]  Jean-Frédéric Gerbeau,et al.  Derivation of viscous Saint-Venant system for laminar shallow water , 2001 .

[6]  Emmanuel Audusse,et al.  A well-balanced positivity preserving second-order scheme for shallow water flows on unstructured meshes , 2005 .

[7]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[8]  Vladimir Zeitlin,et al.  Frontal geostrophic adjustment and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 2. High-resolution numerical simulations , 2004, Journal of Fluid Mechanics.

[9]  M. Bristeau,et al.  3D Free Surface Flows Simulations Using a Multilayer Saint-Venant Model. Comparisons with Navier-Stokes Solutions , 2006 .

[10]  F. Benkhaldoun,et al.  Positivity preserving finite volume Roe: schemes for transport-diffusion equations , 1999 .

[11]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[12]  Marie-Odile Bristeau,et al.  Boundary Conditions for the Shallow Water Equations solved by Kinetic Schemes , 2001 .

[13]  P. Roe,et al.  On Godunov-type methods near low densities , 1991 .

[14]  E. Audusse,et al.  A multilayer Saint-Venant model: Derivation and numerical validation , 2005 .

[15]  Alfredo Bermúdez,et al.  Upwind methods for hyperbolic conservation laws with source terms , 1994 .

[16]  Benoît Perthame,et al.  Kinetic formulation of conservation laws , 2002 .

[17]  B. Perthame,et al.  A kinetic scheme for the Saint-Venant system¶with a source term , 2001 .

[18]  M. Bristeau,et al.  Numerical simulations of 3D free surface flows by a multilayer Saint‐Venant model , 2008 .

[19]  F. Saleri,et al.  A new two-dimensional shallow water model including pressure effects and slow varying bottom topography , 2004 .

[20]  David L. George,et al.  Numerical Approximation of the Nonlinear Shallow Water Equations with Topography and Dry Beds : A Godunov-Type Scheme , 2004 .

[21]  Emmanuel Audusse,et al.  A Multilayer Saint-Venant Model , 2004 .

[22]  V. A. Dougalis,et al.  Numerical Approximation of Blow-Up of Radially Symmetric Solutions of the Nonlinear Schrödinger Equation , 2003, SIAM J. Sci. Comput..

[23]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.