A faster numerical scheme for a coupled system modeling soil erosion and sediment transport

Water Resources Research

Volumn 51, Issue 2, 2015, Pages 987-1005

  Le, M H ^a,b   Cordier, S ^a   Lucas, C ^a   Cerdan, O ^b  

^a UNIVERSITÉ D'ORLÉANS   (France)

^b BRGM   (France)

Author keywords

CFL condition; finite volume method; Hairsine Rose model; numerical stability; shallow water equations; soil erosion

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; EROSION; FINITE VOLUME METHOD; FLOW OF WATER; NUMERICAL METHODS; SEDIMENT TRANSPORT; SEDIMENTATION; SEDIMENTS; SOILS; WATER QUALITY;

CFL CONDITION; MATHEMATICAL PROPERTIES; NUMERICAL APPROXIMATIONS; POSITIVITY PRESERVING SCHEME; SEDIMENT CONCENTRATION; SHALLOW WATER EQUATIONS; SOIL EROSION; TWO-DIMENSIONAL PROBLEM;

EQUATIONS OF MOTION;

CONCENTRATION (COMPOSITION); SEDIMENT TRANSPORT; SHALLOW WATER; SOIL DEGRADATION; SOIL EROSION; WATER DEPTH; WATER FLOW; WATER QUALITY;

EID: 85027931303     PISSN: 00431397     EISSN: 19447973     Source Type: Journal    
DOI: 10.1002/2014WR015690     Document Type: Article

References (57)

1. Aksoy, H., and, M. L. Kavvas, (2005), A review of hillslope and watershed scale erosion and sediment transport models, Catena, 64 (23), 247-271, doi: 10.1016/j.catena.2005.08.008.
2. Audusse, E., and, M.-O. Bristeau, (2003), Transport of pollutant in shallow water: A two time steps kinetic method, Math. Modell. Numer. Anal., 37 (2), 389-416, doi: 10.1051/m2an:2003034.
3. Audusse, E., and, M.-O. Bristeau, (2005), A well-balanced positivity preserving "second-order" scheme for shallow water flows on unstructured meshes, J. Comput. Phys., 206, 311-333, doi: 10.1016/j.jcp.2004.12.016.
4. Audusse, E., F. Bouchut, M.-O. Bristeau, R. Klein, and, B. Perthame, (2004), A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25 (6), 2050-2065, doi: 10.1137/S1064827503431090.
5. Barry, D., G. Sander, S. Jomaa, B. Heng, J.-Y. Parlange, I. Lisle, and, W. Hogarth, (2010), Exact solutions of the Hairsine-Rose precipitation-driven erosion model for a uniform grain-sized soil, J. Hydrol., 389 (34), 399-405, doi: 10.1016/j.jhydrol.2010.06.016.
6. Batten, P., N. Clarke, C. Lambert, and, D. M. Causon, (1997), On the choice of wavespeeds for the HLLC Riemann solver, SIAM J. Sci. Comput., 18 (6), 1553-1570, doi: 10.1137/S1064827593260140.
7. Berthon, C., and, F. Foucher, (2012), Efficient well-balanced hydrostatic upwind schemes for shallow-water equations, J. Comput. Phys., 231 (15), 4993-5015, doi: 10.1016/j.jcp.2012.02.031.
8. Beuselinck, L., G. Govers, A. Steegen, and, T. A. Quine, (1999), Sediment transport by overland flow over an area of net deposition, Hydrol. Processes, 13 (17), 2769-2782.
9. Bouchut, F., (2004), Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, vol. 2, Birkhäuser, Basel, Switzerland, doi: 10.1007/b95203.
10. Bouchut, F., and, T. de Luna, (2010), A subsonic-well-balanced reconstruction scheme for shallow water flows, SIAM J. Numer. Anal., 48 (5), 1733-1758, doi: 10.1137/090758416.
11. Bristeau, M.-O., and, B. Coussin, (2001), Boundary conditions for the shallow water equations solved by kinetic schemes, Tech. Rep. 4282, INRIA. [Available at https://hal.inria.fr/inria-00072305.]
12. Cao, Z., R. Day, and, S. Egashira, (2002), Coupled and decoupled numerical modeling of flow and morphological evolution in alluvial rivers, J. Hydraul. Eng., 128 (3), 306-321, doi: 10.1061/(ASCE)0733-9429(2002)128:3(306).
13. Cao, Z., P. Carling, G. Pender, and, S. Wallis, (2004), Computational dam-break hydraulics over erodible sediment bed, J. Hydraul. Eng., 130, 689-703, doi: 10.1061/(ASCE)0733-9429(2004)130:7(689).
14. Castro, M., J. Gallardo, and, C. Parés, (2006), High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products: Applications to shallow-water systems, Math. Comput., 75 (255), 1103-1134, doi: 10.1090/S0025-5718-06-01851-5.
15. Cerdan, O., et al. (2010), The rate and spatial distribution of soil erosion in Europe: A study based on erosion plot data, Geomorphology, 122, 167-177, doi: 10.1016/j.geomorph.2010.06.011.
16. Cheng, N., (1997), Simplified settling velocity formula for sediment particle, J. Hydraul. Eng., 123 (2), 149-152, doi: 10.1061/(ASCE)0733-9429(1997)123:2(149).
17. Delestre, O., S. Cordier, F. Darboux, and, F. James, (2012), A limitation of the hydrostatic reconstruction technique for shallow water equations, C. R. Math., 350 (13-14), 677-681, doi: 10.1016/j.crma.2012.08.004.
18. Delestre, O., C. Lucas, P.-A. Ksinant, F. Darboux, C. Laguerre, T.-N.-T. Vo, F. James, and, S. Cordier, (2013), SWASHES: A compilation of Shallow Water Analytic solutions for Hydraulic and Environmental Studies, Int. J. Numer. Methods Fluids, 72 (3), 269-300, doi: 10.1002/fld.3741.
19. Duran, A., Q. Liang, and, F. Marche, (2013), On the well-balanced numerical discretization of shallow water equations on unstructured meshes, J. Comput. Phys., 235, 565-586, doi: 10.1016/j.jcp.2012.10.033.
20. Farenhorst, A., and, R. Bryan, (1995), Particle size distribution of sediment transported by shallow flow, Catena, 25 (1-4), 47-62, doi: 10.1016/0341-8162(94)00041-C.
21. Fiener, P., G. Govers, and, K. V. Oost, (2008), Evaluation of a dynamic multi-class sediment transport model in a catchment under soil-conservation agriculture, Earth Surf. Processes Landforms, 33 (11), 1639-1660, doi: 10.1002/esp.1634.
22. Gallouët, T., J.-M. Hérard, and, N. Seguin, (2003), Some approximate Godunov schemes to compute shallow-water equations with topography, Comput. Fluids, 32, 479-513, doi: 10.1016/S0045-7930(02)00011-7.
23. Govers, G., J. Poesen, A. Knapen, G. Gyssels, and, J. Nachtergaele, (2007), Resistance of soils to concentrated flow erosion: A review, Earth Sci. Rev., 80, 75-109, doi: 10.1016/j.earscirev.2006.08.001.
24. Green, W., and, G. Ampt, (1911), Studies on soil physics, J. Agric. Sci., 4, 1-24, doi: 10.1017/S0021859600001441.
25. Greenberg, J. M., and, A.-Y. LeRoux, (1996), A well-balanced scheme for the numerical processing of source terms in hyperbolic equation, SIAM J. Numer. Anal., 33, 1-16, doi: 10.1137/0733001.
26. Hairsine, P., and, C. Rose, (1991), Rainfall detachment and deposition: Sediment transport in the absence of flow-driven processes, Soil Sci. Soc. Am. J., 55 (2), 320-324, doi: 10.2136/sssaj1991.03615995005500020003x.
27. Hairsine, P., and, C. Rose, (1992), Modeling water erosion due to overland flow using physical principles. II: Rill flow, Water Resour. Res., 28, 245-250, doi: 10.1029/91WR02381.
28. Hairsine, P., L. Beuselinck, and, G. Sander, (2002), Sediment transport through an area of net deposition, Water Resour. Res., 38 (6), doi: 10.1029/2001WR000265.
29. Harten, A., P. Lax, and, B. van Leer, (1983), On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1), 35-61, doi: 10.1137/1025002.
30. Heng, B., G. Sander, and, C. Scott, (2009), Modeling overland flow and soil erosion on nonuniform hillslopes: A finite volume scheme, Water Resour. Res., 45, W05423, doi: 10.1029/2008WR007502.
31. Heng, B., G. Sander, A. Armstrong, J. Quinton, J. Chandler, and, C. Scott, (2011), Modeling the dynamics of soil erosion and size-selective sediment transport over nonuniform topography in flume-scale experiments, Water Resour. Res., 47, W02513, doi: 10.1029/2010WR009375.
32. Hervouet, J.-M., (2007), Hydrodynamics of Free Surface Flows: Modelling With the Finite Element Method, John Wiley, Chichester, U. K., doi: 10.1002/9780470319628.
33. Hogarth, W., J.-Y. Parlange, C. Rose, G. Sander, T. Steenhuis, and, A. Barry, (2004), Soil erosion due to rainfall impact with inflow: An analytical solution with spatial and temporal effects, J. Hydrol., 295 (1-4), 140-148, doi: 10.1016/j.jhydrol.2004.03.007.
34. Hou, J., Q. Liang, F. Simons, and, R. Hinkelmann, (2013), A 2D well-balanced shallow flow model for unstructured grids with novel slope source term treatment, Adv. Water Resour., 52, 107-131, doi: 10.1016/j.advwatres.2012.08.003.
35. Kim, J., and, V. Y. Ivanov, (2014), On the nonuniqueness of sediment yield at the catchment scale: The effects of soil antecedent conditions and surface shield, Water Resour. Res., 50, 1025-1045, doi: 10.1002/2013WR014580.
36. Kim, J., V. Y. Ivanov, and, N. D. Katopodes, (2013), Modeling erosion and sedimentation coupled with hydrological and overland flow processes at the watershed scale, Water Resour. Res., 49, 5134-5154, doi: 10.1002/wrcr.20373.
37. LeVeque, R., (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge Univ. Press, Cambridge, U. K.
38. Li, S., and, C. J. Duffy, (2011), Fully coupled approach to modeling shallow water flow, sediment transport, and bed evolution in rivers, Water Resour. Res., 47, W03508, doi: 10.1029/2010WR009751.
39. Liang, Q., and, F. Marche, (2009), Numerical resolution of well-balanced shallow water equations with complex source terms, Adv. Water Resour., 32 (6), 873-884, doi: 10.1016/j.advwatres.2009.02.010.
40. McLachlan, R. I., and, G. R. W. Quispel, (2002), Splitting methods, Acta Numer., 11, 341-434, doi: 10.1017/S0962492902000053.
41. Merritt, W., R. Letcher, and, A. Jakeman, (2003), A review of erosion and sediment transport models, Environ. Model. Software, 18 (8-9), 761-799, doi: 10.1016/S1364-8152(03)00078-1.
42. Mügler, C., O. Planchon, J. Patin, S. Weill, N. Silvera, P. Richard, and, E. Mouche, (2011), Comparison of roughness models to simulate overland flow and tracer transport experiments under simulated rainfall at plot scale, J. Hydrol., 402 (12), 25-40, doi: 10.1016/j.jhydrol.2011.02.032.
43. Murillo, J., P. Garca-Navarro, P. Brufau, and, J. Burguete, (2008), 2D modelling of erosion/deposition processes with suspended load using, J. Hydraul. Res., 46 (1), 99-112, doi: 10.1080/00221686.2008.9521847.
44. Nord, G., and, M. Esteves, (2005), Psem-2d: A physically based model of erosion processes at the plot scale, Water Resour. Res., 41, W08407, doi: 10.1029/2004WR003690.
45. Proffitt, A., C. Rose, and, P. Hairsine, (1991), Rainfall detachment and deposition: Experiments with low slopes and significant water depths, Soil Sci. Soc. Am. J., 55, 325-332, doi: 10.2136/sssaj1991.03615995005500020004x.
46. Quinton, J., G. Govers, K. Van Oost, and, R. Bardgett, (2010), The impact of agricultural soil erosion on biogeochemical cycling, Nat. Geosci., 3, 311-314, doi: 10.1038/ngeo838.
47. Rose, C., B. Yu, H. Ghadiri, H. Asadi, J. Parlange, W. Hogarth, and, J. Hussein, (2007), Dynamic erosion of soil in steady sheet flow, J. Hydrol., 333, 449-458, doi: 10.1016/j.jhydrol.2006.09.016.
48. Sander, G., P. Hairsine, C. Rose, D. Cassidy, J.-Y. Parlange, W. Hogarth, and, I. Lisle, (1996), Unsteady soil erosion model, analytical solutions and comparison with experimental results, J. Hydrol., 178 (1-4), 351-367, doi: 10.1016/0022-1694(95)02810-2.
49. Sander, G., P. Hairsine, L. Beuselinck, and, G. Govers, (2002), Steady state sediment transport through an area of net deposition: Multisize class solutions, Water Resour. Res., 38 (6), doi: 10.1029/2001WR000323.
50. Sander, G. C., J.-Y. Parlange, D. A. Barry, M. B. Parlange, and, W. L. Hogarth, (2007), Limitation of the transport capacity approach in sediment transport modeling, Water Resour. Res., 43, W02403, doi: 10.1029/2006WR005177.
51. Simpson, G., and, S. Castelltort, (2006), Coupled model of surface water flow, sediment transport and morphological evolution, Comput. Geosci., 32, 1600-1614, doi: 10.1016/j.cageo.2006.02.020.
52. Stoker, J. J., (1957), Water Waves: The Mathematical Theory With Applications, Interscience, N. Y.
53. Tatard, L., O. Planchon, J. Wainwright, G. Nord, D. Favis-Mortlock, N. Silvera, O. Ribolzi, M. Esteves, and, C.-H. Huang, (2008), Measurement and modelling of high-resolution flow-velocity data under simulated rainfall on a low-slope sandy soil, J. Hydrol., 348 (1-2), 1-12, doi: 10.1016/j.jhydrol.2007.07.016.
54. Toro, E., (2001), Shock-Capturing Methods for Free-Surface Shallow Flows, 326 pp., John Wiley, N. Y.
55. van Leer, B., (1979), Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, J. Comput. Phys., 32 (1), 101-136, doi: 10.1016/0021-9991(79)90145-1.
56. Van Oost, K., L. Beuselinck, P. B. Hairsine, and, G. Govers, (2004), Spatial evaluation of a multi-class sediment transport and deposition model, Earth Surf. Processes Landforms, 29 (8), 1027-1044, doi: 10.1002/esp.1089.
57. Woolhiser, D., and, J. Liggett, (1967), Unsteady, one-dimensional flow over a plane-The rising hydrograph, Water Resour. Res., 3, 753-771, doi: 10.1029/WR003i003p00753.