Efficient numerical schemes for viscoplastic avalanches. Part 1: The 1D case

Authors:
Enrique D. Fernández-Nieto;José M. Gallardo;Paul Vigneaux
Affiliations:
Departamento de Matemática Aplicada I, Universidad de Sevilla, E.T.S. Arquitectura, Avda, Reina Mercedes, s/n, 41012 Sevilla, Spain;Departamento de Análisis Matemático, Universidad de Málaga, F. Ciencias, Campus Teatinos S/N, Spain;Unitée de Mathématiques Pures et Appliquées, Ecole Normale Supérieure de Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France
Venue:
Journal of Computational Physics
Year:
2014

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Abstract

This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non-trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated with the yield threshold: finite-volume schemes are used together with duality methods (namely Augmented Lagrangian and Bermudez-Moreno) to discretize the problem. To be able to accurately simulate the stopping behavior of the avalanche, new schemes need to be designed, involving the classical notion of well-balancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. We derived such schemes and numerical experiments are presented to show their performances.