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

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 an original method showing that such 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 spread 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 movement is a little bit quicker, shallow water theory for non Newtonian flows may be tried to handle with for instance assuming Navier type boundary condition at the bottom. We start from the variational inequality for 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 condition that relates 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 the spacial discretization with the augmented Lagrangian method to treat the associated optimization problem. Finally, we present various numerical tests.
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Didier Bresch, Enrique D. Fernandez - Nieto, Ioan R. Ionescu, Paul Vigneaux. Augmented Lagrangian Method and Compressible Visco-Plastic Flows : Applications to Shallow Dense Avalanches. Fursikov, Andrei V., Galdi, Giovanni P., Pukhnachev, Vladislav V. New Directions in Mathematical Fluid Mechanics, Birkhäuser, pp.57-89, 2010, Advances in Mathematical Fluid Mechanics, 978-3-0346-0151-1. ⟨hal-00327369⟩

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