F. Table, Test 1. Errors and order of convergence for Augmented Lagrangian ? µ. Cells L ? Error Order L 2 Error

F. Table, 7: Test 1. Errors and order of convergence for Bermúdez

C. Acary-robert, E. D. Fernández-nieto, G. Narbona-reina, and P. Vigneaux, A Well-balanced Finite Volume-Augmented Lagrangian Method for an Integrated Herschel-Bulkley Model, Journal of Scientific Computing, vol.41, issue.11, pp.608-641, 2012.
DOI : 10.1029/2005WR004475
URL : https://hal.archives-ouvertes.fr/hal-00709491

C. Ancey, Plasticity and geophysical flows: A review, Journal of Non-Newtonian Fluid Mechanics, vol.142, issue.1-3, pp.4-35, 2007.
DOI : 10.1016/j.jnnfm.2006.05.005

N. J. Balmforth, A. S. Burbidge, R. V. Craster, J. Salzig, and A. Shen, Visco-plastic models of isothermal lava domes, Journal of Fluid Mechanics, vol.403, issue.4, pp.37-65, 2000.
DOI : 10.1017/S0022112099006916

A. Bermúdez and C. Moreno, Duality methods for solving variational inequalities, Computers & Mathematics with Applications, vol.7, issue.1, pp.43-58, 1981.
DOI : 10.1016/0898-1221(81)90006-7

A. Bermúdez and M. E. Vázquez-cendón, Upwind methods for hyperbolic conservation laws with source terms, Computers & Fluids, vol.23, issue.8, pp.1049-1071, 1994.
DOI : 10.1016/0045-7930(94)90004-3

D. Boffi, Finite element approximation of eigenvalue problems, Acta Numerica, vol.19, pp.1-120, 2010.
DOI : 10.1017/S0962492910000012

F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Birkhäuser, 2004.

D. Bresch, E. D. Fernandez-nieto, I. R. Ionescu, and P. Vigneaux, Augmented Lagrangian Method and Compressible Visco-plastic Flows: Applications to Shallow Dense Avalanches, New Directions in Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, pp.57-89, 2010.
DOI : 10.1007/978-3-0346-0152-8_4
URL : https://hal.archives-ouvertes.fr/hal-00327369

H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, p.31, 1973.

M. J. Castro, J. M. Gonzáles-vida, and C. Parés, NUMERICAL TREATMENT OF WET/DRY FRONTS IN SHALLOW FLOWS WITH A MODIFIED ROE SCHEME, Mathematical Models and Methods in Applied Sciences, vol.440, issue.06, pp.897-931, 2006.
DOI : 10.1006/jcph.1998.6127

M. J. Castro, A. M. Ferreiro-ferreiro, J. A. García-rodríguez, J. M. González-vida, J. Macías et al., The numerical treatment of wet/dry fronts in shallow flows: application to one-layer and two-layer systems, Mathematical and Computer Modelling, vol.42, issue.3-4, pp.3-4419, 2005.
DOI : 10.1016/j.mcm.2004.01.016

T. Chacón, M. J. Castro, E. D. Fernández-nieto, and C. Parés, On well-balanced finite volume methods for non-conservative nonhomogeneous hyperbolic systems, SIAM J. Sci. Comput, vol.29, issue.3 2, pp.1093-1126, 2007.

F. Delbos, J. Ch, R. Gilbert, D. Glowinski, and . Sinoquet, Constrained optimization in seismic reflection tomography: a Gauss-Newton augmented Lagrangian approach, Geophysical Journal International, vol.164, issue.3, pp.670-684, 2006.
DOI : 10.1111/j.1365-246X.2005.02729.x

J. D. Dent and T. E. Lang, A biviscous modified Bingham model of snow avalanche motion, Ann. Glaciol, vol.4, issue.2, p.4246, 1983.

G. Duvaut and J. Lions, Inequalities in mechanics and physics, 1976.

I. Ekeland and R. Témam, Convex analysis and variational problems, Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), vol.28, pp.31-32, 1999.
DOI : 10.1137/1.9781611971088

E. D. Fernández-nieto, J. M. Gallardo, and P. Vigneaux, Efficient numerical schemes for viscoplastic avalanches. Part 2: the 2D case, Journal of Computational Physics, vol.5, p.22, 2017.
DOI : 10.1016/j.jcp.2017.09.054

E. D. Fernández-nieto, P. Noble, and J. Vila, Shallow Water equations for Non-Newtonian fluids, Journal of Non-Newtonian Fluid Mechanics, vol.165, issue.13-14, pp.712-732, 1314.
DOI : 10.1016/j.jnnfm.2010.03.008

M. Fortin and R. Glowinski, Augmented Lagrangian methods: applications to the numerical solution of boundary-value problems, p.17, 1983.

M. José, C. Gallardo, M. Parés, and . Castro, A generalized duality method for solving variational inequalities. Applications to some nonlinear Dirichlet problems, Numer. Math, vol.100, issue.2, pp.259-291, 2005.

J. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin, Dyn. Syst., Ser. B, vol.1, issue.1 4, pp.89-102, 2001.
URL : https://hal.archives-ouvertes.fr/hal-00691701

R. Glowinski and P. L. Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), vol.9, issue.29, p.30, 1989.
DOI : 10.1137/1.9781611970838

R. Glowinski and A. Wachs, On the Numerical Simulation of Viscoplastic Fluid Flow, Numerical Methods for Non-Newtonian Fluids, pp.483-717, 2011.
DOI : 10.1016/B978-0-444-53047-9.00006-X

J. M. Greenberg and A. Roux, A Well-Balanced Scheme for the Numerical Processing of Source Terms in Hyperbolic Equations, SIAM Journal on Numerical Analysis, vol.33, issue.1, pp.1-16, 1996.
DOI : 10.1137/0733001

R. R. Huilgol and Z. You, Application of the augmented Lagrangian method to steady pipe flows of Bingham, Casson and Herschel???Bulkley fluids, Journal of Non-Newtonian Fluid Mechanics, vol.128, issue.2-3, pp.126-143, 2005.
DOI : 10.1016/j.jnnfm.2005.04.004

I. R. Ionescu, Augmented Lagrangian for shallow viscoplastic flow with topography, Journal of Computational Physics, vol.242, issue.2, pp.544-560, 2013.
DOI : 10.1016/j.jcp.2013.02.029

K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Society for Industrial and Applied Mathematics, issue.6, 2008.
DOI : 10.1137/1.9780898718614

R. J. Leveque, Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods: The Quasi-Steady Wave-Propagation Algorithm, Journal of Computational Physics, vol.146, issue.1, pp.346-365, 1998.
DOI : 10.1006/jcph.1998.6058

D. Kirill, M. A. Nikitin, K. M. Olshanskii, Y. V. Terekhov, and . Vassilevski, A numerical method for the simulation of free surface flows of viscoplastic fluid in 3D, Journal of Computational Mathematics, vol.29, issue.6 1, pp.605-622, 2011.

T. C. Papanastasiou, Flows of Materials with Yield, Journal of Rheology, vol.31, issue.5, pp.385-404, 1987.
DOI : 10.1122/1.549926

C. Parés, M. Castro, and J. Macías, On the convergence of the Berm??dez-Moreno algorithm with constant parameters, Numerische Mathematik, vol.92, issue.1, pp.113-128, 2002.
DOI : 10.1007/s002110100352

C. Parés, J. Macías, and M. Castro, Duality methods with an automatic choice of parameters Application to shallow water equations in conservative form, Numerische Mathematik, vol.89, issue.1, pp.161-189, 2001.
DOI : 10.1007/PL00005461

P. L. Roe, Upwind differencing schemes for hyperbolic conservation laws with source terms, Nonlinear hyperbolic problems, pp.41-51, 1986.
DOI : 10.1007/BFb0008660

N. Roquet and P. Saramito, An adaptive finite element method for Bingham fluid flows around a cylinder, Computer Methods in Applied Mechanics and Engineering, vol.192, issue.31-32, pp.31-323317, 2003.
DOI : 10.1016/S0045-7825(03)00262-7

E. F. Toro, Shock-capturing methods for free-surface shallow flows, 2001.

G. Vinay, A. Wachs, and J. Agassant, Numerical simulation of weakly compressible Bingham flows: The restart of pipeline flows of waxy crude oils, Journal of Non-Newtonian Fluid Mechanics, vol.136, issue.2-3, pp.93-105, 2006.
DOI : 10.1016/j.jnnfm.2006.03.003
URL : https://hal.archives-ouvertes.fr/hal-00673899

D. Vola, F. Babik, and J. Latché, On a numerical strategy to compute gravity currents of non-Newtonian fluids, Journal of Computational Physics, vol.201, issue.2, pp.397-420, 2004.
DOI : 10.1016/j.jcp.2004.05.019

D. Vola, L. Boscardin, and J. C. Latché, Laminar unsteady flows of Bingham fluids: a numerical strategy and some benchmark results, Journal of Computational Physics, vol.187, issue.2, pp.441-456, 2003.
DOI : 10.1016/S0021-9991(03)00118-9