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.
Mathematics Subject Classification (2000)
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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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