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Augmented Lagrangian Method and Compressible Visco-plastic Flows: Applications to Shallow Dense Avalanches

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New Directions in Mathematical Fluid Mechanics

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

In this paper we propose a well-balanced finite volume/augmented Lagrangian method for compressible visco-plastic models focusing on a compressible Bingham type system with applications to dense avalanches. For the sake of completeness we also present a method showing that such a system may be derived for a shallow flow of a rigid-viscoplastic incompressible fluid, namely for incompressible Bingham type fluid with free surface. When the fluid is relatively shallow and spreads slowly, lubrication-style asymptotic approximations can be used to build reduced models for the spreading dynamics, see for instance [N.J. Balmforth et al., J. Fluid Mech (2002)]. When the motion is a little bit quicker, shallow water theory for non-Newtonian flows may be applied, for instance assuming a Navier type boundary condition at the bottom. We start from the variational inequality for an incompressible Bingham fluid and derive a shallow water type system. In the case where Bingham number and viscosity are set to zero we obtain the classical Shallow Water or Saint-Venant equations obtained for instance in [J.F. Gerbeau, B. Perthame, DCDS (2001)]. For numerical purposes, we focus on the one-dimensional in space model: We study associated static solutions with sufficient conditions that relate the slope of the bottom with the Bingham number and domain dimensions. We also propose a well-balanced finite volume/augmented Lagrangian method. It combines well-balanced finite volume schemes for spatial discretization with the augmented Lagrangian method to treat the associated optimization problem. Finally, we present various numerical tests.

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Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques UMR 5127 CNRS, Univ. Savoie, F-73376, Le Bourget du Lac, France

    D. Bresch

  2. Dpto. Matemática Aplicada I, E.T.S. Arquitectura, U. Sevilla, Avda. Reina Mercedes n. 2, E-41012, Sevilla, Spain

    E. D. Fernández-Nieto

  3. Laboratoire P.M.T.M., Institut Galilée #x2014; Université Paris 13, Avenue Jean-Baptiste Clément, F-93430, Villetaneuse, France

    I. R. Ionescu

  4. Unité de Mathématiques Pures et Appliquées, ENS de Lyon, 46, allée d’Italie, F-69364, Lyon Cedex 07, France

    P. Vigneaux

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  1. D. Bresch
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  2. E. D. Fernández-Nieto
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  3. I. R. Ionescu
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  4. P. Vigneaux
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Editors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, Vorob’evy Gory, 119991, Moscow, Russia

    Andrei V. Fursikov

  2. Department of Mechanical Engineering and Materials Science, University of Pittsburgh, 630 Benedum Engineering Hall, Pittsburgh, PA, 15261, USA

    Giovanni P. Galdi

  3. Lavrentyev Institute of Hydrodynamics, Lavrentyev prospect 15, 630090, Novosibirsk, Russia

    Vladislav V. Pukhnachev

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Dedicated to the memory of Professor Alexandre V. Kazhikov

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Bresch, D., Fernández-Nieto, E.D., Ionescu, I.R., Vigneaux, P. (2009). Augmented Lagrangian Method and Compressible Visco-plastic Flows: Applications to Shallow Dense Avalanches. In: Fursikov, A.V., Galdi, G.P., Pukhnachev, V.V. (eds) New Directions in Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0152-8_4

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