Parallelization of a relaxation scheme modelling the bedload transport of sediments in shallow water flow. (English. French summary) Zbl 1329.76221
Summary: We are interested in numerical simulations for bedload erosion processes. We present a relaxation solver that we apply to moving dunes test cases in one and two dimensions. In particular we retrieve the so-called anti-dune process that is well described in the experiments. In order to be able to run 2D test cases with reasonable CPU time, we also describe and apply a parallelization procedure by using domain decomposition based on the classical MPI library.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
| 65Y05 | Parallel numerical computation |
| 86A05 | Hydrology, hydrography, oceanography |
References:
| [1] | E. Audusse and M.-O. Bristeau. A well-balanced positivity preserving ”second-order” scheme for shallow water flows on unstructured meshes. Journal of Computational Physics, 206:311–333, 2005. · Zbl 1087.76072 |
| [2] | E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, and B. Perthame. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput., 25(6):2050–2065, 2004. · Zbl 1133.65308 |
| [3] | E. Audusse, C. Berthon, C. Chalons, O. Delestre, N. Goutal, M. Jodeau, J. Sainte-Marie, J. Giesselmann, and G. Sadaka. Sediment transport modelling : Relaxation schemes for Saint-Venant - Exner and three layer models. ESAIM: Proc., 38:78–98, 2012. URL http://dx.doi.org/10.1051/proc/201238005. · Zbl 1407.86003 |
| [4] | F. Benkhaldoun, S. Sahmim, and M. Sea\"{}ıd. A two-dimensional finite volume morphodynamic model on unstructured triangular grids. International Journal for Numerical Methods in Fluids, 2009. |
| [5] | F. Benkhaldoun, S. Sahmim, and M. Sea\"{}ıd. Solution of the sediment transport equations using a finite volume method based on sign matrix. SIAM J. Sci. Comp, 31:2866–2889, 2009. · Zbl 1201.35021 |
| [6] | A. Berm\'{}udez and M.E. V\'{}azquez. Upwind methods for hyperbolic conservation laws with source terms. Computers & Fluids, 23(8):1049–1071, 1994. ISSN 0045-7930.URL http://www.sciencedirect.com/ · Zbl 0816.76052 |
| [7] | A. Berm\'{}udez, A. Dervieux, J.-A. Desideri, and M. Elena V\'{}azquez. Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes. Computer Methods in Applied Mechanics and Engineering, 155(1-2):49–72, 1998. ISSN 0045-7825.URL http://www.sciencedirect. com/science/article/B6V29-3WN739F-4/2/3120ee07ea5c43b3318fbfb6c05d2dde. · Zbl 0961.76047 |
| [8] | C. Berthon, S. Cordier, O. Delestre, and M.-H. Le. An analytical solution of the shallow water system coupled to the Exner equation. Comptes Rendus Mathematique, 350(3–4):183–186, 2012. ISSN 1631-073X. URL http://www.sciencedirect.com/science/article/pii/S1631073X12000088. · Zbl 1235.35218 |
| [9] | M. Bilanceri, F. Beux, I. Elmahi, H. Guillard, and M.V. Salvetti. Linearised implicit time-advancing applied to sediment transport simulations. Technical Report 7492, INRIA, 2010. · Zbl 1246.76066 |
| [10] | D. Bresch, M.J.C. D\'{}ıaz, E.D. Fern\'{}andez-Nieto, A.M. Ferreiro, and A. Mangeney. High order finite volume methods applied to sediment transport and submarine avalanches. In Sylvie Benzoni-Gavage and Denis Serre, editors, Hyperbolic Problems: Theory, Numerics, Applications, pages 247–258. Springer Berlin Heidelberg, 2008. ISBN 978-3-540-75711-5. URL http://dx.doi.org/10.1007/978-3-540-75712-2_20. · Zbl 1388.76167 |
| [11] | I. Brugeas. Utilisation de MPI en d\'{}ecomposition de domaine, March 1996. URL http://www.idris.fr/ data/publications/mpi.ps. |
| [12] | V. Caleffi, A. Valiani, and A. Bernini. High-order balanced CWENO scheme for movable bed shallow water equations. Advances in Water Ressources, 30:730–741, 2007. |
| [13] | B. Camenen and M. Larson. A general formula for non-cohesive bed load sediment transport. Estuarine Coastal and Shelf Science, 63:249–260, 2005. |
| [14] | C. Camporeale and L. Ridolfi. Nonnormality and transient behavior of the de Saint-Venant-Exner equations. Water Resources Research, 45(8):n/a–n/a, 2009. ISSN 1944-7973.URL http://dx.doi.org/10.1029/ 2008WR007587. |
| [15] | A. Canestrelli, M. Dumbser, A. Siviglia, and E. Toro. Well-balanced high-order centred schemes for nonconservative hyperbolic systems. applications to shallow water equations with fixed and mobile bed. Advances in Water Ressources, 32:834–844, 2009. |
| [16] | A. Canestrelli, M. Dumbser, A. Siviglia, and E. Toro. Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed. Advances in Water Ressources, 33:291–303, 2010. |
| [17] | M.J. Castro, A. Pardo, and C. Par‘es. Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique. Mathematical Models and Methods in Applied Sciences, 17(12):2065–2113, 2007. · Zbl 1137.76038 |
| [18] | M.J. Castro-D\'{}ıaz, E.D. Fern\'{}andez-Nieto, and A.M. Ferreiro. Sediment transport models in shallow water equations and numerical approach by high order finite volume methods. Computers & Fluids, 37 (3):299–316, March 2008. ISSN 0045-7930.URL http://www.sciencedirect.com/science/article/ B6V26-4PSK8R5-1/2/9043410cc90abc43b410c70660a4efd6. · Zbl 1237.76082 |
| [19] | M.J. Castro D\'{}ıaz, E.D. Fern\'{}andez-Nieto, A.M. Ferreiro, and C. Par\'{}es.Two-dimensional sediment transport models in shallow water equations. a second order finite volume approach on unstructured meshes. Computer Methods in Applied Mechanics and Engineering, 198(33-36):2520–2538, 2009. ISSN 0045-7825.URL http://www.sciencedirect.com/science/article/B6V29-4VWB151-1/2/ b1cf68d800ba7d81d182133a1945bc35. |
| [20] | F. Charru and H. Mouilleron-Arnould. Instability of a bed of particles sheared by a viscous flow. Journal of Fluid Mechanics, 452:303–323, 1 2002. ISSN 1469-7645.URL http://dx.doi.org/10.1017/ S0022112001006747. · Zbl 1059.76023 |
| [21] | M. Colombini. Revisiting the linear theory of sand dune formation. Journal of Fluid Mechanics, 502:1–16, 2004. ISSN 1469-7645. · Zbl 1057.76021 |
| [22] | M. Colombini and A. Stocchino. Finite-amplitude river dunes. Journal of Fluid Mechanics, 611:283–306, 8 2008. ISSN 1469-7645. URL http://dx.doi.org/10.1017/S0022112008002814. · Zbl 1151.76484 |
| [23] | S. Cordier, M.-H. Le, and T. Morales de Luna. Bedload transport in shallow water models: Why splitting (may) fail, how hyperbolicity (can) help. Advances in Water Resources, 34(8):980 – 989, 2011. ESAIM: PROCEEDINGS93 |
| [24] | G. Dal Maso, P.G. Lefloch, and F. Murat. Definition and weak stability of nonconservative products. J. Math. Pures Appl. (9), 74(6):483–548, 1995. ISSN 0021-7824. · Zbl 0853.35068 |
| [25] | J. De Dieu Zabsonr\'{}e, C. Lucas, and E. Fern\'{}andez-Nieto. An energetically consistent viscous sedimentation model. Mathematical Models and Methods in Applied Sciences, 19(03):477–499, 2009. URL http://www. worldscientific.com/doi/abs/10.1142/S0218202509003504. · Zbl 1178.35308 |
| [26] | O. Delestre, C. Lucas, P.-A. Ksinant, F. Darboux, C. Laguerre, T.-N.-Tuoi Vo, F. James, and S. Cordier. SWASHES: a compilation of Shallow Water Analytic Solutions for Hydraulic and Environmental Studies. International Journal for Numerical Methods in Fluids, 72(3):269–300, 2013. ISSN 1097-0363.URL http://dx.doi.org/10.1002/fld.3741. |
| [27] | A.I. Delis and I. Papoglou. Relaxation approximation to bedload sediment transport. Journal of Computation and Applied Mathematics, 213:521–546, 2008. · Zbl 1144.76036 |
| [28] | O. Devauchelle, L. Malverti, E. Lajeunesse, P.-Y. Lagree, C. Josserand, and K.-D. Nguyen Thu-Lam. Stability of bedforms in laminar flows with free surface: from bars to ripples. Journal of Fluid Mechanics, 642:329–348. ISSN 1469-7645. URL http://dx.doi.org/10.1017/S0022112009991790. · Zbl 1183.76731 |
| [29] | Olivier Devauchelle, Christophe Josserand, Pierre-Yves Lagr\'{}ee, and St\'{}ephane Zaleski. Morphodynamic modeling of erodible laminar channels. Phys. Rev. E, 76:056318, Nov 2007. URL http://link.aps.org/ doi/10.1103/PhysRevE.76.056318. |
| [30] | H. El-Rewini and M. Abd-El-Barr. Advanced Computer Architecture and Parallel Processing (Wiley Series on Parallel and Distributed Computing). Wiley-Interscience, 2005. ISBN 0471467405. |
| [31] | George Em Karniadakis and Rober M. Kirby II. Parallel Scientific Computing in C++ and MPI. Cambridge University Press, 2003. · Zbl 1027.68023 |
| [32] | F.M. Exner.\"{}Uber die wechselwirkung zwischen wasser und geschiebe in fl\"{}ussen. Sitzungsber., Akad. Wissenschaften, pt. IIa:Bd. 134, 1925. |
| [33] | M. Garcia. Sedimentation engineering: Processes, measurements, modeling and practice. ASCE Manuals and Reports on Engineering Practice No. 110, 2008. |
| [34] | J.-F. Gerbeau and B. Perthame. Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B, 1(1):89–102, 2001. ISSN 1531-3492. · Zbl 0997.76023 |
| [35] | A.J. Grass. Sediment transport by waves and currents. SERC London Cent. Mar. Technol, Report No. FL29, 1981. |
| [36] | J. M. Greenberg and A.-Y. LeRoux. A well-balanced scheme for the numerical processing of source terms in hyperbolic equation. SIAM Journal on Numerical Analysis, 33:1–16, 1996. |
| [37] | H. P. Guy, D. B. Simons, and E. V. Richardson. Summary of alluvial channel data from flume experiments. Geol. Survay Prof. Paper 462-I, 1966. |
| [38] | L.-H. Huang and Y.-L. Chiang. The formation of dunes, antidunes, and rapidly damping waves in alluvial channels. International Journal for Numerical and Analytical Methods in Geomechanics, 25(7):675–690, 2001. ISSN 1096-9853. URL http://dx.doi.org/10.1002/nag.147. · Zbl 1052.74556 |
| [39] | J. Hudson. Numerical techniques for morphodynamic modelling. PhD thesis, University of Reading, October 2001. |
| [40] | J. Hudson and P. K. Sweby. A high-resolution scheme for the equations governing 2d bed-load sediment transport. International Journal for Numerical Methods in Fluids, 47(10-11):1085–1091, 2005. ISSN 10970363. URL http://dx.doi.org/10.1002/fld.853. · Zbl 1064.76072 |
| [41] | J. Hudson and P.K. Sweby. Formulations for numerically approximating hyperbolic systems governing sediment transport. J. Sci. Comput., 19(1-3):225–252, December 2003. ISSN 0885-7474.URL http: //dx.doi.org/10.1023/A:1025304008907. · Zbl 1081.76572 |
| [42] | J. Hudson, J. Damgaard, N. Dodd, T. Chesher, and A. Cooper. Numerical approaches for 1d morphodynamic modelling. Coastal Engineering, 52:691–707, 2005. |
| [43] | S. Jin. A steady-state capturing method for hyperbolic systems with geometrical source terms. M2AN, 35 (4):631–645, July 2001. URL http://dx.doi.org/10.1051/m2an:2001130. · Zbl 1001.35083 |
| [44] | S. Jin and X. Wen. efficient method for computing hyperbolic systems with geometrical source terms having concentrations. J. Comput. Math., 22:230–249, 2004. · Zbl 1119.65373 |
| [45] | S. Jin and X. Wen. Two interface type numerical methods for computing hyperbolic systems with geometrical source terms having concentrations. SIAM J. Sci. Comp., 26:2079–2101, 2005. · Zbl 1083.35062 |
| [46] | J.-F. Kennedy. The formation of sediment ripples, dunes, and antidunes. Annual Review of Fluid Mechanics, 1(1):147–168, 1969. URL http://www.annualreviews.org/doi/abs/10.1146/annurev.fl.01.010169. 001051. |
| [47] | John F. Kennedy. The mechanics of dunes and antidunes in erodible-bed channels. Journal of Fluid Mechanics, 16, 1963. . · Zbl 0122.43602 |
| [48] | K. K. J. Kouakou and P.-Y. Lagr\'{}ee.Evolution of a model dune in a shear flow.Eur J Mech B Fluids, 25(3):348–359, October 2005.ISSN 0997-7546.URL http://pubget.com/paper/pgtmp_ 4e1e901b36d5a552b68e97d95e82086a. · Zbl 1130.76337 |
| [49] | P.-Y. Lagr\'{}ee. A triple deck model of ripple formation and evolution. Physics of Fluids, 15(8):2355–2368, 2003. URL http://link.aip.org/link/?PHF/15/2355/1. · Zbl 1186.76302 |
| [50] | M.-H. Le. Mod\'{}elisation multi-\'{}echelle et simulation num\'{}erique de l’\'{}erosion des sols de la parcelle au bassin versant. PhD thesis, Universit\'{}e d’Orl\'{}eans (in French), available from TEL: http://tel.archivesouvertes.fr/tel-00780648, November 2012. |
| [51] | R.J. LeVeque. Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm.Journal of Computational Physics, 146(1):346–365, 1998. ISSN 0021-9991.URL http://www.sciencedirect.com/science/article/B6WHY-45J58TW-22/2/ 72a8af8c9e23f63b0df9484475f7e2df. · Zbl 0931.76059 |
| [52] | Q. Liang. A coupled morphodynamic model for applications involving wetting and drying. J. Hydrodynamics, 23(3):273–281, 2011. |
| [53] | Q. Liang and F. Marche. Numerical resolution of well-balanced shallow water equations with complex source terms. Advances in Water Resources, 32(6):873–884, 2009. ISSN 0309-1708.URL http://www. sciencedirect.com/science/article/B6VCF-4VR9FHT-3/2/5218868dcfded7d34380666f23000855. |
| [54] | R. Fernandez Luque and R. van Beek. Erosion and transport of bedload sediment. J. Hydraul. Res., 14 (2):127–144, 1976. |
| [55] | E. Meyer-Peter and R. M\"{}uller. Formulas for bed-load transport. In 2nd meeting IAHSR, Stockholm, Sweden, pages 1–26, 1948. |
| [56] | J. Murillo and P. Garc\'{}ıa-Navarro. An Exner-based coupled model for two-dimensional transient flow over erodible bed. Journal of Computational Physics, 229:8704–8732, 2010. · Zbl 1282.76131 |
| [57] | P. Nielsen. Coastal Bottom Boundary Layers and Sediment Transport. World Scientific Pub Co Inc, August 1992. ISBN 9810204736. |
| [58] | P. Pacheco. An Introduction to Parallel Programming. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1st edition, 2011. ISBN 9780123742605. |
| [59] | G. Parker. Sediment inertia as cause of river antidunes. J. Hydraul. Div. ASCE, 101:211–221, 1975. |
| [60] | A. Shields. Application of similarity principles and turbulence research to bedload movement. Mitteilunger der Preussischen Versuchsanstalt f\"{}ur Wasserbau und Schiffbau, 26:5–24, 1936. |
| [61] | S. Soares-Fraz\tilde{}ao and Y. Zech. HLLC scheme with novel wave-speed estimators appropriate for twodimensional shallow-water flow on erodible bed. Int. J. Num. Meth. Fluids, 66(8):1019–1036, 2010. · Zbl 1285.76025 |
| [62] | Y. Zech, S. Soares-Frazao, B. Spinewine, and N. Le Grelle. Dam-break induced sediment movement: Experimental approaches and numerical modelling. Journal of Hydraulic Research, 46:176–190, 2008. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
