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**A well-balanced approach for flows over mobile-bed with high sediment-transport.**
*(English)*
Zbl 1158.86302

Summary: We deal with the numerical computation of one-dimensional, unsteady, free-surface flows over mobile-bed. We focus on flows characterized by high concentration of sediments and strong interaction between flow and bottom dynamics, as in hyper-concentrated- and debris-flows. These features are fully considered in the adopted system of equations. Challenging in its numerical approximation is the preservation of the coupling and the treatment of a non-conservative flux in the momentum equation. In order to devise a new Godunov-type approach, we analyzed in detail the Riemann problem associated with the mobile-bed phenomena and the peculiar features of its wave relations. The scheme we developed is based on two supports: well-balanced treatment of the variable updating at the new time-level and flux evaluation by three-wave approximations of the intercell Riemann-problem that, without any split, embody the effect of the non-conservative term. The properties of the new numerical strategy (named AWB) are assessed by comparison with exact solutions of Riemann problems, built by handling an inverse technique. Finally, AWB has been applied to cases of practical interest, where wave interaction and friction effects makes the flow more complex. The obtained results point out that the new method is able to predict faithfully the overall behaviour of the solution and of any type of waves. The use of AWB, in this one-dimensional frame, is therefore fostered in representing rapid transients in river/torrent flows with movable bed.

### MSC:

86-08 | Computational methods for problems pertaining to geophysics |

76B07 | Free-surface potential flows for incompressible inviscid fluids |

76M20 | Finite difference methods applied to problems in fluid mechanics |

86A05 | Hydrology, hydrography, oceanography |

### Keywords:

well-balanced scheme; Riemann solvers; non-conservative term; mobile-bed models; sediment transport
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\textit{G. Rosatti} and \textit{L. Fraccarollo}, J. Comput. Phys. 220, No. 1, 312--338 (2006; Zbl 1158.86302)

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### References:

[1] | Alcrudo, F.; Benkhaldoun, F., Exact solutions to the Riemann problem of the shallow water equations with a bottom step, Comput. Fluids, 30, 643-671 (2001) · Zbl 1048.76008 |

[2] | Andrianov, N., Performance of numerical methods on the non-unique solution to the Riemann problem for the shallow water equations, Int. J. Numer. Meth. Fluids, 47, 825-831 (2005) · Zbl 1134.76402 |

[3] | Andrianov, N.; Warneckle, G., On the solution to the Riemann problem for the compressible duct flow, SIAM J. Appl. Math., 64, 3, 878-901 (2004) · Zbl 1065.35191 |

[4] | Armanini, A.; Capart, H.; Fraccarollo, L.; Larcher, M., Rheological stratification in experimental free-surface flows of granular liquid mixtures, J. Fluid Mech., 532, 269-319 (2005) · Zbl 1125.76301 |

[5] | Ashida, K.; Michiue, M., Study on hydraulic resistance and bedload rate in alluvial streams, Trans. Jpn. Soc. Civil Eng., 206, 59-69 (1972) |

[6] | Bermudez, A.; Vázquez, M. E., Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids, 23, 1049-1071 (1994) · Zbl 0816.76052 |

[7] | Cao, Z.; Pender, G.; Wallis, S.; Carling, P., Computational dam-break hydraulics over erodible sediment bed, J. Hydraulic Eng., 130, 7, 689-703 (2004) |

[8] | Capart, H.; Young, D. L., Formation of a jump by the dam-break wave over a granular bed, J. Fluid Mech., 372, 165-187 (1998) · Zbl 0941.86001 |

[9] | C.E. Castro, E.F. Toro, A Riemann solver and upwind methods for a two-phase flow model in non-conservative form, Isaac Newton Institute for Mathematical Sciences, Preprint Series, no. NI05001-NPA, 2005.; C.E. Castro, E.F. Toro, A Riemann solver and upwind methods for a two-phase flow model in non-conservative form, Isaac Newton Institute for Mathematical Sciences, Preprint Series, no. NI05001-NPA, 2005. · Zbl 1086.76046 |

[10] | Chinnayya, A.; LeRoux, A.-Y.; Seguin, N., A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: The resonance phenomenon, Int. J. Finite Volumes (2004) · Zbl 1490.65167 |

[11] | Črnjarić-Žic, N.; Vuković, S.; Sopta, L., Extension of ENO and WENO schemes to one-dimensional sediment transport equations, Comput. Fluids, 33, 31-56 (2004) · Zbl 1165.76359 |

[12] | Dai, W.; Woodward, P. R., An iterative Riemann solver for relativistic hydrodynamics, SIAM J. Sci. Comput., 14, 4, 982-995 (1997) · Zbl 0892.35008 |

[13] | Fraccarollo, L.; Capart, H., Riemann wave description of erosional dam-break flows, J. Fluid Mech., 461, 183-228 (2002) · Zbl 1142.76344 |

[14] | Fraccarollo, L.; Capart, H.; Zech, Y., A Godunov method for the computation of erosional shallow water transient, Int. J. Numer. Meth. Fluids, 41, 9, 951-976 (2003) · Zbl 1028.76028 |

[15] | Godunov, S. K., Finite difference method for numerical computation of discontinuous solution of the equations of fluid dynamics, Mat. Sb., 47, 271-300 (1959) · Zbl 0171.46204 |

[16] | Greenberg, J. M.; LeRoux, A.-Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33, 1-16 (1996) · Zbl 0876.65064 |

[17] | Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, Uniformly high order accuracy essentially non-oscillatory schemes III, J. Comput. Phys., 71, 231-303 (1987) · Zbl 0652.65067 |

[18] | Harten, A.; Lax, P. D.; Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 35-61 (1983) · Zbl 0565.65051 |

[19] | Isaacson, E.; Temple, B., Convergence of the 2×2 Godunov method for general resonant nonlinear balance law, SIAM J. Appl. Math., 55-3, 625-640 (1995) · Zbl 0838.35075 |

[20] | LeVeque, R. J., Numerical Methods for Conservation Laws (1992), Birkhäuser: Birkhäuser Basel · Zbl 0847.65053 |

[21] | E. Meyer-Peter, R. Müller, Formulas for bed-load transport, in: Proc. of 2nd Meeting IAHSR, Stockolm, Sweden, 1948. pp. 1-26.; E. Meyer-Peter, R. Müller, Formulas for bed-load transport, in: Proc. of 2nd Meeting IAHSR, Stockolm, Sweden, 1948. pp. 1-26. |

[22] | Morris, P. H.; Williams, D. J., Relative celerities of mobile bed flows with finite solids concentrations, J. Hydraulic Eng., 122, 311-315 (1996) |

[23] | van Rijn, L., Sedimentation of dredged channels by currents and waves, J. Waterway Port Ocean Eng., 2, 5 (1986) |

[24] | Rosatti, G.; Fraccarollo, L.; Armanini, A., Behaviour of small perturbations in 1d mobile-bed models, (Della Morte, R.; Greco, M.; Carravetta, A., RiverFlow 2004 (2004), Balkema: Balkema The Netherlands) |

[25] | Sumer, B. M.; Kozakiewicz, A.; Fredsøe, J.; Deigaard, R., Velocity and concentration profiles in sheet-flow layer of movable bed, J. Hydraulic Eng., 122, 549-558 (1996) |

[26] | Takahashi, T., Debris Flow (1991), Balkema: Balkema The Netherlands |

[27] | Titarev, V. A.; Toro, E. F., ADER: arbitrary high order Godunov approach, J. Sci. Comput., 17, 609-618 (2002) · Zbl 1024.76028 |

[28] | Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics (1999), Springer: Springer Berlin · Zbl 0923.76004 |

[29] | E.F. Toro, Multi-stage predictor-corrector fluxes for hyperbolic equations, Isaac Newton Institute for Mathematical Sciences, Preprint Series NI03037-NPA, University of Cambridge, UK, 2003.; E.F. Toro, Multi-stage predictor-corrector fluxes for hyperbolic equations, Isaac Newton Institute for Mathematical Sciences, Preprint Series NI03037-NPA, University of Cambridge, UK, 2003. |

[30] | Weisstein, E. W., CRC Concise Encyclopedia of Mathematics/Eric W. Weisstein (1998), CRC Press: CRC Press Boca Raton, FL |

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