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

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 Bermúdez-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.
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Enrique D. Fernandez-Nieto, J.M. Gallardo, Paul Vigneaux. Efficient numerical schemes for viscoplastic avalanches. Part 1: the 1D case. Journal of Computational Physics, Elsevier, 2014, 264, pp.55-90. ⟨10.1016/j.jcp.2014.01.026⟩. ⟨hal-01011373⟩

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