We present a hybrid finite-volume-particle numerical method for computing the transport of a passive pollutant by a flow. The flow is modeled by the one- and two-dimensional Saint-Venant system of shallow water equations and the pollutant propagation is described by a transport equation. This paper is an extension of our previous work [Chertock, Kurganov and Petrova, J. Sci. Comput. (to appear)], where the one-dimensional finite-volume-particle method has been proposed. The core idea behind the finite-volume-particle method is to use different schemes for the flow and pollution computations: the shallow water equations are numerically integrated using a finite-volume scheme, while the transport equation is solved by a particle method. This way the specific advantages of each scheme are utilized at the right place. A special attention is given to the recovery of the point values of the numerical solution from its particle distribution. The reconstruction is obtained using a dual equation for the pollutant concentration. This results in a significantly enhanced resolution of the computed solution and also makes it much easier to extend the finite-volume-particle method to the two-dimensional case.
Mots clés : shallow water equations, transport of passive pollutant, finite-volume schemes, particle method
@article{M2AN_2004__38_6_1071_0,
author = {Chertock, Alina and Kurganov, Alexander},
title = {On a hybrid finite-volume-particle method},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {1071--1091},
publisher = {EDP-Sciences},
volume = {38},
number = {6},
year = {2004},
doi = {10.1051/m2an:2004051},
mrnumber = {2108945},
zbl = {1077.65091},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an:2004051/}
}
TY - JOUR AU - Chertock, Alina AU - Kurganov, Alexander TI - On a hybrid finite-volume-particle method JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2004 SP - 1071 EP - 1091 VL - 38 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2004051/ DO - 10.1051/m2an:2004051 LA - en ID - M2AN_2004__38_6_1071_0 ER -
%0 Journal Article %A Chertock, Alina %A Kurganov, Alexander %T On a hybrid finite-volume-particle method %J ESAIM: Modélisation mathématique et analyse numérique %D 2004 %P 1071-1091 %V 38 %N 6 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2004051/ %R 10.1051/m2an:2004051 %G en %F M2AN_2004__38_6_1071_0
Chertock, Alina; Kurganov, Alexander. On a hybrid finite-volume-particle method. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 6, pp. 1071-1091. doi : 10.1051/m2an:2004051. http://archive.numdam.org/articles/10.1051/m2an:2004051/
[1] ,,, and, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050-2065. | Zbl
[2] and, Transport of pollutant in shallow water. A two time steps kinetic method. ESAIM: M2AN 37 (2003) 389-416. | Numdam | Zbl
[3] ,, and, A wave propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 24 (2002) 955-978. | Zbl
[4] and, Transport of pollutant in shallow water using kinetic schemes. CEMRACS, Orsay (electronic), ESAIM Proc., Paris. Soc. Math. Appl. Indust. 10 (1999) 9-21.
[5] , and, Finite-volume-particle methods for models of transport of pollutant in shallow water. J. Sci. Comput. (to appear). | MR | Zbl
[6] and, Optimal approximations of transport equations by particle and pseudoparticle methods. SIAM J. Math. Anal. 32 (2000) 616-636. | Zbl
[7] , and, Nonlinear filters for efficient shock computation. Math. Comp. 52 (1989) 509-537. | Zbl
[8] , Differential equations with discontinuous right-hand side. (Russian). Mat. Sb. (N.S.) 51 (1960) 99-128. | Zbl
[9] , Differential equations with discontinuous right-hand side. AMS Transl. 42 (1964) 199-231. | Zbl
[10] , Differential equations with discontinuous right-hand side, Translated from the Russian. Kluwer Academic Publishers Group, Dordrecht. Math. Appl. (Soviet Series) 18 (1988). | MR | Zbl
[11] , and, Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32 (2003) 479-513. | Zbl
[12] and, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 89-102. | Zbl
[13] , and, High order time discretization methods with the strong stability property. SIAM Rev. 43 (2001) 89-112. | Zbl
[14] and, Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397-425. | Numdam | Zbl
[15] and, On the reduction of numerical dissipation in central-upwind schemes (in preparation).
[16] , and, Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21 (2001) 707-740. | Zbl
[17] and, Central schemes and contact discontinuities. ESAIM: M2AN 34 (2000) 1259-1275. | Numdam | Zbl
[18] and, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241-282. | Zbl
[19] , Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 101-136. | Zbl
[20] and, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-463. | Zbl
[21] and, A kinetic scheme for the Saint-Venant system with a source. Calcolo 38 (2001) 201-231. | Zbl
[22] , An analysis of particle methods, in Numerical methods in fluid dynamics (Como, 1983). Lect. Notes Math. 1127 (1985) 243-324. | Zbl
[23] , Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit. C. R. Acad. Sci. Paris 73 (1871) 147-154. | JFM
[24] , High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21 (1984) 995-1011. | Zbl
Cité par Sources :
