S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune et al., PETSc users manual, 2018.

S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune et al., , 2018.

N. J. Balmforth and R. R. Kerswell, Granular collapse in two dimensions, Journal of Fluid Mechanics, vol.538, pp.399-428, 2005.

T. Barker and J. M. Gray, Partial regularisation of the incompressible µ(I)-rheology for granular flow, J. Fluid Mech, vol.828, pp.5-32, 2017.

T. Barker, D. G. Schaeffer, P. Bohórquez, and J. M. Gray, Well-posed and ill-posed behaviour of the µ(I)-rheology for granular flow, J. Fluid Mech, vol.779, pp.794-818, 2015.

N. Brodu, R. Delannay, . P. Valance, and . Richard, New patterns in high-speed granular flows, J. Fluid Mech, vol.769, pp.218-228, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01126756

R. Chalayer, L. Chupin, and T. Dubois, A bi-projection method for incompressible bingham flows with variable density, viscosity, and yield stress, SIAM J. Numer. Anal, vol.56, issue.4, pp.2461-2483, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01803711

Y. C. Chang, T. Y. Hou, B. Merriman, and S. Osher, A level set formulation of eulerian interface capturing methods for incompressible fluid flows, J. Comput. Phys, vol.124, 1996.

A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp, vol.22, pp.745-762, 1968.

L. Chupin and T. Dubois, A bi-projection method for Bingham type flows, Comput. Math. Appl, vol.72, issue.5, pp.1263-1286, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01166406

L. Chupin and J. Mathé, Existence theorem for homogeneous incompressible Navier-Stokes equation with variable rheology, Eur. J. Mech. B-Fluid, vol.61, pp.135-143, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01381040

G. B. Crosta, S. Imposimato, and D. Roddeman, Numerical modeling of 2-d granular step collapse on erodible and nonerodible surface, J. Geophys. Res. Earth Surface, vol.114, issue.F3, 2009.

R. Delannay, . Valance, A. Mangeney, O. Roche, and . Richard, Granular and particle-laden flows: from laboratory experiments to field observations, J. Phys. D Appl. Phys, vol.50, issue.5, p.53001, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01481019

G. Della-rocca and G. Blanquart, Level set reinitialization at a contact line, J. Comput. Phys, vol.265, pp.34-49, 2014.

. Gdr-midi, On dense granular flows, Eur. Phys. J, vol.14, pp.341-365, 2004.

L. Girolami, V. Hergault, G. Vinay, and A. Wachs, A three-dimensional discrete-grain model for the simulation of dam-break rectangular collapses: comparison between numerical results and experiments, Granul. Matter, vol.14, issue.3, pp.381-392, 2012.
URL : https://hal.archives-ouvertes.fr/hal-02171451

R. Glowinsky, J. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, 1981.

S. Gottlieb and C. Shu, Total variation diminishing Runge-Kutta schemes, Math. Comp, vol.67, issue.221, pp.73-85, 1998.

J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng, vol.195, issue.44, pp.6011-6045, 2006.

V. Gueugneau, K. Kelfoun, O. Roche, and L. Chupin, Effects of pore pressure in pyroclastic flows: Numerical simulation and experimental validation, Geophys. Res. Lett, vol.44, issue.5, pp.2194-2202, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01625001

F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, vol.12, issue.8, pp.2182-2189, 1965.

I. R. Ionescu, A. Mangeney, F. Bouchut, and O. Roche, Viscoplastic modeling of granular column collapse with pressure-dependent rheology, J. Non-Newt. Fluid, vol.219, pp.1-18, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01080456

G. Jiang and D. Peng, Weighted eno schemes for hamilton-jacobi equations, SIAM J. Sci. Comput, vol.21, issue.6, 2000.

P. Jop, Y. Forterre, and O. Pouliquen, A constitutive law for dense granular flows, Nature, vol.441, pp.727-730, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01432178

R. R. Kerswell, Dam break with coulomb friction: A model for granular slumping?, Phys. Fluids, vol.17, issue.5, p.57101, 2005.

L. Lacaze and R. R. Kerswell, Axisymmetric granular collapse: A transient 3d flow test of viscoplasticity, Phys. Rev. Lett, vol.102, p.108305, 2009.

L. Lacaze, J. C. Phillips, and R. R. Kerswell, Planar collapse of a granular column: Experiments and discrete element simulations, Phys. Fluids, vol.20, issue.6, p.63302, 2008.

P. Lagrée, L. Staron, and S. Popinet, The granular column collapse as a continuum: Validity of a two-dimensional Navier-Stokes model with a µ(I)-rheology, J. Fluid Mech, vol.686, pp.378-408, 2011.

E. Larrieu, L. Staron, and E. J. Hinch, Raining into shallow water as a description of the collapse of a column of grains, J. Fluid Mech, vol.554, pp.259-270, 2006.

Y. Liu, N. J. Balmforth, S. Hormozi, and D. R. Hewitt, Two-dimensional viscoplastic dambreaks, J. Non-Newt. Fluid, vol.238, pp.65-79, 2016.

A. Mangeney, O. Roche, O. Hungr, N. Mangold, G. Faccanoni et al., Erosion and mobility in granular collapse over sloping beds, J. Geophys. Res. Earth Surface, vol.115, issue.F3, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00521671

A. Mangeney-castelnau, F. Bouchut, J. P. Vilotte, E. Lajeunesse, A. Aubertin et al., On the use of saint venant equations to simulate the spreading of a granular mass, J. Geophys. Res. Solid Earth, vol.110, 2005.

N. Martin, I. R. Ionescu, A. Mangeney, F. Bouchut, and M. Farin, Continuum viscoplastic simulation of a granular column collapse on large slopes: µ(I) rheology and lateral wall effects, Phys. Fluids, vol.29, issue.1, p.13301, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01204716

C. Min, On reinitializing level set functions, J. Comput. Phys, vol.229, 2010.

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, vol.153, 2003.

O. Roche, Depositional processes and gas pore pressure in pyroclastic flows: an experimental perspective, B. Volcanol, vol.74, issue.8, pp.1807-1820, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00793602

O. Roche, S. Montserrat, Y. Niño, and A. Tamburrino, Pore fluid pressure and internal kinematics of gravitational laboratory air-particle flows: Insights into the emplacement dynamics of pyroclastic flows, J. Geophys. Res. Solid Earth, vol.115, issue.B9, p.9206, 2010.

D. G. Schaeffer, Instability in the evolution equations describing incompressible granular flow, J. Differ. Equations, vol.66, issue.1, pp.19-50, 1987.

D. G. Schaeffer and E. B. Pitman, Ill-posedness in three-dimensional plastic flow, Commun. Pure Appl. Math, vol.41, issue.7, pp.879-890, 1988.

C. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shockcapturing schemes, J. Comput. Phys, vol.77, issue.2, pp.439-471, 1988.

M. Sussman, E. Fatemi, P. Smereka, and S. Osher, An improved level set method for incompressible two-phase flows, Comp. Fluids, vol.27, pp.5-6, 1998.

M. Sussman, P. Smereka, and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys, vol.114, 1994.

M. Sussman, K. M. Smith, M. Y. Hussaini, M. Otha, and R. Zhi-wei, A sharp interface method for incompressible two-phase flows, J. Comput. Phys, vol.221, issue.2, 2007.

R. Temam, Sur l'approximation de la solution deséquations de Navier-Stokes par la méthode des pas fractionnaires, II. Arch. Rational Mech. Anal, vol.33, pp.377-385, 1969.