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

Journal of Computational Physics

Volumn 264, Issue , 2014, Pages 55-90

  Fernández Nieto, Enrique D ^a   Gallardo, José M ^b   Vigneaux, Paul ^c  

^a UNIVERSITY OF SEVILLE   (Spain)

^b UNIVERSITY OF MÁLAGA   (Spain)

^c UNIVERSITÉ DE LYON   (France)

Author keywords

Bingham; Finite volume; Shallow water; Variational inequality; Viscoplastic; Well balanced

Indexed keywords

CONSTRAINED OPTIMIZATION; NEWTONIAN LIQUIDS; VARIATIONAL TECHNIQUES;

1-D CASE; BINGHAM; EFFICIENT NUMERICAL SCHEMES; FINITE-VOLUME; NUMERICAL RESOLUTION; SHALLOW WATERS; VARIATIONAL INEQUALITIES; VISCOPLASTIC; VISCOPLASTIC FLOWS; WELL BALANCED;

YIELD STRESS;

EID: 84893800327     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2014.01.026     Document Type: Article

References (37)

1. Acary-Robert C., Fernández-Nieto E., Narbona-Reina G., Vigneaux P. A well-balanced finite volume-augmented lagrangian method for an integrated Herschel-Bulkley model. J. Sci. Comput. 2012, 53:608-641.
2. Ancey C. Plasticity and geophysical flows: A review. J. Non-Newton. Fluid Mech. 2007, 142:4-35.
3. Balmforth N.J., Burbidge A.S., Craster R.V., Salzig J., Shen A. Visco-plastic models of isothermal lava domes. J. Fluid Mech. 2000, 403:37-65.
4. Bermúdez A., Moreno C. Duality methods for solving variational inequalities. Comput. Math. Appl. 1981, 7:43-58.
5. Bermúdez A., Vázquez Cendón M.E. Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 1994, 23:1049-1071.
6. Boffi D. Finite element approximation of eigenvalue problems. Acta Numer. 2010, 19:1-120.
7. Bouchut F. Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources 2004, Birkhäuser.
8. Bresch D., Fernandez-Nieto E.D., Ionescu I.R., Vigneaux P. Augmented Lagrangian method and compressible visco-plastic flows: Applications to shallow dense avalanches. New Directions in Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics 2010, 57-89. Birkhäuser, Basel. G.P. Galdi (Ed.).
9. Brézis H. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert 1973, North-Holland Publishing Co.
10. Castro M., Ferreiro Ferreiro A., García-Rodríguez J., González-Vida J., Macías J., Parés C., Elena Vázquez-Cendón M. The numerical treatment of wet/dry fronts in shallow flows: Application to one-layer and two-layer systems. Math. Comput. Model. 2005, 42(3-4):419-439.
11. Castro M.J., Gonzáles-Vida J.M., Parés C. Numerical treatment of wet/dry fronts in shallow flows with a modified roe scheme. Math. Models Methods Appl. Sci. 2006, 16(6):897-931.
12. Chacón T., Castro M.J., Fernández-Nieto E.D., Parés C. On well-balanced finite volume methods for non-conservative non-homogeneous hyperbolic systems. SIAM J. Sci. Comput. 2007, 29(3):1093-1126.
13. Delbos F., Gilbert J.C., Glowinski R., Sinoquet D. Constrained optimization in seismic reflection tomography: a Gauss-Newton augmented Lagrangian approach. Geophys. J. Int. 2006, 164(3):670-684.
14. Dent J., Lang T. A biviscous modified Bingham model of snow avalanche motion. Ann. Glaciol. 1983, 4:42-46.
15. Duvaut G., Lions J.L. Inequalities in Mechanics and Physics 1976, Springer-Verlag.
16. Ekeland I., Témam R. Convex Analysis and Variational Problems. Classics in Applied Mathematics 1999, vol. 28. Society for Industrial and Applied Mathematics (SIAM).
17. Fernández-Nieto E.D., Noble P., Vila J.P. Shallow water equations for non-newtonian fluids. J. Non-Newton. Fluid Mech. 2010, 165(13-14):712-732.
18. Fortin M., Glowinski R. Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems 1983, North-Holland.
19. Gallardo J.M., Parés C., Castro M. A generalized duality method for solving variational inequalities. Applications to some nonlinear Dirichlet problems. Numer. Math. 2005, 100:259-291.
20. Gerbeau J.F., Perthame B. Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst., Ser. B 2001, 1:89-102.
21. Glowinski R., Le Tallec P. Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM Studies in Applied Mathematics 1989, vol. 9. Society for Industrial and Applied Mathematics (SIAM).
22. Glowinski R., Wachs A. On the numerical simulation of viscoplastic fluid flow. Handbook of Numerical Analysis 2011, vol. 16:483-717. Elsevier. R. Glowinski, J. Xu (Eds.).
23. Greenberg J.M., Le Roux A.Y. A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 1996, 33:1-16.
24. Huilgol R.R., You Z. Application of the augmented Lagrangian method to steady pipe flows of Bingham, Casson and Herschel-Bulkley fluids. J. Non-Newton. Fluid Mech. 2005, 128(2-3):126-143.
25. Ionescu I.R. Augmented lagrangian for shallow viscoplastic flow with topography. J. Comput. Phys. 2013, 242:544-560.
26. Ito K., Kunisch K. Lagrange Multiplier Approach to Variational Problems and Applications 2008, Society for Industrial and Applied Mathematics.
27. LeVeque R.J. Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 1998, 146:346-365.
28. Nikitin K.D., Olshanskii M.A., Terekhov K.M., Vassilevski Y.V. A numerical method for the simulation of free surface flows of viscoplastic fluid in 3D. J. Comput. Math. 2011, 29:605-622.
29. Papanastasiou T. Flows of materials with yield. J. Rheol. 1987, 385-404.
30. Parés C., Castro M., Macías J. On the convergence of the Bermúdez-Moreno algorithm with constant parameters. Numer. Math. 2002, 92(1):113-128.
31. Parés C., Macías J., Castro M. Duality methods with an automatic choice of parameters. Application to shallow water equations in conservative form. Numer. Math. 2001, 89:161-189.
32. Roe P.L. Upwind differencing schemes for hyperbolic conservation laws with source terms. Lecture Notes in Math. 1987, vol. 1270:41-51. Springer, Berlin. C. Carraso (Ed.).
33. Roquet N., Saramito P. An adaptive finite element method for Bingham fluid flows around a cylinder. Comput. Methods Appl. Mech. Eng. 2003, 192(31-32):3317-3341.
34. Toro E.F. Shock-Capturing Methods for Free-Surface Shallow Flows 2001, John Wiley & Sons.
35. Vinay G., Wachs A., Agassant J.F. Numerical simulation of weakly compressible Bingham flows: The restart of pipeline flows of waxy crude oils. J. Non-Newton. Fluid Mech. 2006, 136(2-3):93-105.
36. Vola D., Babik F., Latché J.C. On a numerical strategy to compute gravity currents of non-Newtonian fluids. J. Comput. Phys. 2004, 201:397-420.
37. Vola D., Boscardin L., Latché J.C. Laminar unsteady flows of Bingham fluids: a numerical strategy and some benchmark results. J. Comput. Phys. 2003, 187:441-456.