Accurate arrested states for finite-volumes schemes on a shallow Bingham model

Paul Vigneaux (1, 2)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 NUMED - Numerical Medicine

We start here from a prototype of a 2D Shallow Water Bingham model as derived in Bresch et al (2010), see also Ionescu (2013). For such integrated models, Finite Volume methods are known to be very efficient. In this talk, we present a new 2D well-balanced scheme for such discretizations which is able to compute accurately the arrested states of avalanches of Bingham fluids on real topographies and in the presence of wet/dry fronts. This well-balanced approach needs to be coordinated with the duality method used to solve the viscoplastic nature of the model (leading to variational inequalities). The Bingham law is here solved unregularized via the Augmented Lagrangian or the Bermudez-Moreno (BM) methods: for both methods a rigorous study of the numerical cost is performed and an a priori estimation of the optimal duality parameter is given for the BM (extending the 1D case treated in Fernandez-Nieto et al (2014)). It is shown that this a priori estimate allows to be close to the shortest computation times. To illustrate the ability of these schemes to handle the numerical difficulties encountered in geophysical applications (complex DEM topographies, long space domains and long time scales), we present in particular a simulation of an avalanche in the Taconnaz path (Chamonix, Mont-Blanc). This is a joint work with E.D. Fernandez-Nieto and J. M. Gallardo. /-/ References: D. Bresch, E.D. Fernandez-Nieto, I. Ionescu, P. Vigneaux. Augmented Lagrangian Method and Compressible Visco-Plastic Flows : Applications to Shallow Dense Avalanches. Advances in Mathematical Fluid Mechanics, 2010, pp. 57-89. I. Ionescu. Augmented Lagrangian for shallow viscoplastic flow with topography. Journal of Computational Physics, 2013, Vol. 242, pp 544-560. E.D. Fernandez-Nieto, J. M. Gallardo, P. Vigneaux. Efficient numerical schemes for viscoplastic avalanches. Part 1: the 1D case. Journal of Computational Physics, 2014, Vol. 264, pp 55-90. E.D. Fernandez-Nieto, J. M. Gallardo, P. Vigneaux. Efficient numerical schemes for viscoplastic avalanches. Part 2: the 2D case. Journal of Computational Physics, accepted 2017. https://hal.archives-ouvertes.fr/hal-01593148 /-/ Acknowledgements : This work has been partially supported by CNRS through the interdisciplinary program InFIniti 2017.

Mots clés: Taconnaz; Viscoplastic; Shallow Water; Finite Volume; Well-Balanced; Variational inequality; Bingham

Domaines: Analyse numérique [math.NA]; Mécanique des fluides [physics.class-ph]

Paul Vigneaux. Accurate arrested states for finite-volumes schemes on a shallow Bingham model. 7th Viscoplastic Fluids Workshop: From Theory to Applications (VPF7), Oct 2017, Rotorua, New Zealand. ⟨hal-01630047⟩