Hybrid central-upwind schemes for numerical resolution of two-phase flows
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 2, p. 253-273

In this paper we present a methodology for constructing accurate and efficient hybrid central-upwind (HCU) type schemes for the numerical resolution of a two-fluid model commonly used by the nuclear and petroleum industry. Particularly, we propose a method which does not make use of any information about the eigenstructure of the jacobian matrix of the model. The two-fluid model possesses a highly nonlinear pressure law. From the mass conservation equations we develop an evolution equation which describes how pressure evolves in time. By applying a quasi-staggered Lax-Friedrichs type discretization for this pressure equation together with a Modified Lax-Friedrich type discretization of the convective terms, we obtain a central type scheme which allows to cope with the nonlinearity (nonlinear pressure waves) of the two-fluid model in a robust manner. Then, in order to obtain an accurate resolution of mass fronts, we employ a modification of the convective mass fluxes by hybridizing the central type mass flux components with upwind type components. This hybridization is based on a splitting of the mass fluxes into components corresponding to the pressure and volume fraction variables, recovering an accurate resolution of a contact discontinuity. In the numerical simulations, the resulting HCU scheme gives results comparable to an approximate Riemann solver while being superior in efficiency. Furthermore, the HCU scheme yields better robustness than other popular Riemann-free upwind schemes.

DOI : https://doi.org/10.1051/m2an:2005011
Classification:  35L65,  65M12,  76N10,  76T10
Keywords: two-phase flow, two-fluid model, hyperbolic system of conservation laws, central discretization, upwind discretization, pressure evolution equation, hybrid scheme
@article{M2AN_2005__39_2_253_0,
     author = {Evje, Steinar and Fl\aa tten, Tore},
     title = {Hybrid central-upwind schemes for numerical resolution of two-phase flows},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {2},
     year = {2005},
     pages = {253-273},
     doi = {10.1051/m2an:2005011},
     zbl = {1130.76057},
     mrnumber = {2143949},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2005__39_2_253_0}
}
Evje, Steinar; Flåtten, Tore. Hybrid central-upwind schemes for numerical resolution of two-phase flows. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 2, pp. 253-273. doi : 10.1051/m2an:2005011. http://www.numdam.org/item/M2AN_2005__39_2_253_0/

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