The surface gradient method for the treatment of source terms in the shallow-water equations.

*(English)*Zbl 1074.86500Summary: A novel scheme has been developed for data reconstruction within a Godunov-type method for solving the shallow-water equations with source terms. In contrast to conventional data reconstruction methods based on conservative variables, the water surface level is chosen as the basis for data reconstruction. This provides accurate values of the conservative variables at cell interfaces so that the fluxes can be accurately calculated with a Riemann solver. The main advantages are: (1) a simple centered discretization is used for the source terms; (2) the scheme is no more complicated than the conventional method for the homogeneous terms; (3) small perturbations in the water surface elevation can be accurately predicted; and (4) the method is generally suitable for both steady and unsteady shallow-water problems. The accuracy of the scheme has been verified by recourse to both steady and unsteady flow problems. Excellent agreement has been obtained between the numerical predictions and analytical solutions. The results indicate that the new scheme is accurate, simple, efficient, and robust.

##### MSC:

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

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

86A05 | Hydrology, hydrography, oceanography |

##### Keywords:

source terms; shallow-water equations; data reconstruction; high-resolution method; Godunov method; MUSCL scheme
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\textit{J. G. Zhou} et al., J. Comput. Phys. 168, No. 1, 1--25 (2001; Zbl 1074.86500)

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##### References:

[1] | Alcrudo, F.; Garcia-Navarro, P., A high-resolution Godunov-type scheme in finite volumes for the 2d shallow-water equations, Int. J. numer. methods fluids, 16, 489, (1993) · Zbl 0766.76067 |

[2] | Mingham, C.G.; Causon, D.M., High-resolution finite-volume method for shallow water flows, J. hydraul.-eng. ASCE, 124, 605, (1998) · Zbl 0931.76051 |

[3] | Hu, K.; Mingham, C.G.; Causon, D.M., A bore-capturing finite volume method for open-channel flows, Int. J. numer. methods fluids, 28, 1241, (1998) · Zbl 0931.76051 |

[4] | Causon, D.M.; Ingram, D.M.; Mingham, C.G.; Yang, G.; Pearson, R.V., Calculation of shallow water flows using a Cartesian cut cell approach, Adv. water resour., 23, 545, (2000) |

[5] | Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1997) · Zbl 0888.76001 |

[6] | LeVeque, R.J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, J. comput. phys., 146, 346, (1998) · Zbl 0931.76059 |

[7] | Bermudez, A.; Vzquez, M.E., Upwind methods for hyperbolic conservation laws with source terms, Comput. fluids, 23, 1049, (1994) · Zbl 0816.76052 |

[8] | Vázquez-Cendón, M.E., Improved treatment of source terms in upwind schemes for shallow water equations in channels with irregular geometry, J. comput. phys., 148, 497, (1999) · Zbl 0931.76055 |

[9] | LeVeque, R.J., Numerical methods for conservation laws, (1990) · Zbl 0682.76053 |

[10] | van Leer, B., On the relation between the upwind-differencing schemes of Godunov, enguist – osher and roe, SIAM J. sci. stat. comput., 5, 1, (1985) · Zbl 0547.65065 |

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

[12] | Toro, E.F., Riemann problems and the WAF method for solving two-dimensional shallow water equations, Philos.trans. R. soc. London ser. A, 338, 43, (1992) · Zbl 0747.76027 |

[13] | Fraccarollo, L.; Toro, E.F., Experimental and numerical assessment of the shallow water model for two-dimensional dam-break type problems, J. hydraul. res., 33, 843, (1995) |

[14] | Bermudez, A.; Dervieux, A.; Desideri, J.; Vázquez, M.E., Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes, Comput. methods appl. mech. eng., 155, 49, (1998) · Zbl 0961.76047 |

[15] | Goutal, N.; Maurel, F., Proceedings of the 2nd workshop on dam-break wave simulation, (1997) |

[16] | Chow, V.T., Open-channel hydraulics, (1959) |

[17] | Zhou, J.G.; Stansby, P.K., 2D shallow water model for hydraulic jump, Int. J. numer. methods fluids, 29, 375, (1999) · Zbl 0938.76057 |

[18] | Gharangik, A.M.; Chaudhry, M.H., Numerical simulation of hydraulic jump, J. hydraul. eng. ASCE, 117, 1195, (1991) |

[19] | Molls, T.; Chaudhry, M.H., Depth-averaged open-channel flow model, J. hydraul. eng. ASCE, 121, 453, (1995) |

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