A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport.

*(English)*Zbl 1268.76037Summary: The goal of this paper is to obtain a well-balanced, stable, fast, and robust HLLC-type approximate Riemann solver (cf. [A. Harten, P. D. Lax and B. van Leer, SIAM Rev. 25, 35–61 (1983; Zbl 0565.65051)] for a hyperbolic nonconservative PDE system arising in a turbidity current model. The main difficulties come from the nonconservative nature of the system. A general strategy to derive simple approximate Riemann solvers for nonconservative systems is introduced, which is applied to the turbidity current model to obtain two different HLLC solvers. Some results concerning the non-negativity preserving property of the corresponding numerical methods are presented. The numerical results provided by the two HLLC solvers are compared between them and also with those obtained with a Roe-type method in a number of 1d and 2d test problems. This comparison shows that, while the quality of the numerical solutions is comparable, the computational cost of the HLLC solvers is lower, as only some partial information of the eigenstructure of the matrix system is needed.

##### MSC:

76M12 | Finite volume methods applied to problems in fluid mechanics |

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

76N99 | Compressible fluids and gas dynamics |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |