ESAIM: Mathematical Modelling and Numerical Analysis

Table of Contents - September 2008 - Volume 42 - Issue05

Research Article

A Roe-type scheme for two-phase shallow granular flows over variable topography

Pelanti, Maricaa1, Bouchut, Françoisa2 and Mangeney, Annea3a4

a1 Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France. Marica.Pelanti@ens.fr

a2 CNRS and Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France. Francois.Bouchut@ens.fr

a3 Équipe de Sismologie, Institut de Physique du Globe de Paris, 4 place Jussieu, 75252 Paris Cedex 05, France. mangeney@ipgp.jussieu.fr

a4 Institute for Nonlinear Science, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0402, USA.

Abstract

We study a depth-averaged model of gravity-driven flows made of solid grains and fluid, moving over variable basal surface. In particular, we are interested in applications to geophysical flows such as avalanches and debris flows, which typically contain both solid material and interstitial fluid. The model system consists of mass and momentum balance equations for the solid and fluid components, coupled together by both conservative and non-conservative terms involving the derivatives of the unknowns, and by interphase drag source terms. The system is hyperbolic at least when the difference between solid and fluid velocities is sufficiently small. We solve numerically the one-dimensional model equations by a high-resolution finite volume scheme based on a Roe-type Riemann solver. Well-balancing of topography source terms is obtained via a technique that includes these contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problems of perturbed steady flows over non-flat bottom surface that show the efficient modeling of disturbances of equilibrium conditions.

(Received November 9 2007)

(Online publication July 30 2008)

Key Words:

  • Granular flows;
  • two-phase flows;
  • thin layer approximation;
  • non-conservative systems;
  • numerical model;
  • finite volume schemes;
  • Riemann solvers;
  • well-balanced schemes.

Mathematics Subject Classification:

  • 65M99;
  • 76T25
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