On a hybrid finite-volume-particle method
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 6, pp. 1071-1091.

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.

DOI : https://doi.org/10.1051/m2an:2004051
Classification : 34A36,  35L67,  35Q35,  65M99
Mots clés : shallow water equations, transport of passive pollutant, finite-volume schemes, particle method
     author = {Chertock, Alina and Kurganov, Alexander},
     title = {On a hybrid finite-volume-particle method},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - 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},
     zbl = {1077.65091},
     mrnumber = {2108945},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2004051/}
Chertock, Alina; Kurganov, Alexander. On a hybrid finite-volume-particle method. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 6, pp. 1071-1091. doi : 10.1051/m2an:2004051. http://www.numdam.org/articles/10.1051/m2an:2004051/

[1] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050-2065. | Zbl 1133.65308

[2] E. Audusse and M.-O. Bristeau, Transport of pollutant in shallow water. A two time steps kinetic method. ESAIM: M2AN 37 (2003) 389-416. | Numdam | Zbl 1137.65392

[3] D.S. Bale, R.J. Leveque, S. Mitran and J.A. Rossmanith, A wave propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 24 (2002) 955-978. | Zbl 1034.65068

[4] M.-O. Bristeau and B. Perthame, Transport of pollutant in shallow water using kinetic schemes. CEMRACS, Orsay (electronic), ESAIM Proc., Paris. Soc. Math. Appl. Indust. 10 (1999) 9-21.

[5] A. Chertock, A. Kurganov and G. Petrova, Finite-volume-particle methods for models of transport of pollutant in shallow water. J. Sci. Comput. (to appear). | MR 2285775 | Zbl 1101.76036

[6] A. Cohen and B. Perthame, Optimal approximations of transport equations by particle and pseudoparticle methods. SIAM J. Math. Anal. 32 (2000) 616-636. | Zbl 0972.65058

[7] B. Engquist, P. Lötstedt and B. Sjögreen, Nonlinear filters for efficient shock computation. Math. Comp. 52 (1989) 509-537. | Zbl 0667.65073

[8] A.F. Filippov, Differential equations with discontinuous right-hand side. (Russian). Mat. Sb. (N.S.) 51 (1960) 99-128. | Zbl 0138.32204

[9] A.F. Filippov, Differential equations with discontinuous right-hand side. AMS Transl. 42 (1964) 199-231. | Zbl 0148.33002

[10] A.F. Filippov, Differential equations with discontinuous right-hand side, Translated from the Russian. Kluwer Academic Publishers Group, Dordrecht. Math. Appl. (Soviet Series) 18 (1988). | MR 1028776 | Zbl 0664.34001

[11] T. Gallouët, J.-M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32 (2003) 479-513. | Zbl 1084.76540

[12] J.F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 89-102. | Zbl 0997.76023

[13] S. Gottlieb, C.-W. Shu and E. Tadmor, High order time discretization methods with the strong stability property. SIAM Rev. 43 (2001) 89-112. | Zbl 0967.65098

[14] A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397-425. | Numdam | Zbl 1137.65398

[15] A. Kurganov and C.-T. Lin, On the reduction of numerical dissipation in central-upwind schemes (in preparation).

[16] A. Kurganov, S. Noelle and G. Petrova, Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21 (2001) 707-740. | Zbl 0998.65091

[17] A. Kurganov and G. Petrova, Central schemes and contact discontinuities. ESAIM: M2AN 34 (2000) 1259-1275. | Numdam | Zbl 0972.65055

[18] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241-282. | Zbl 0987.65085

[19] B. Van Leer, Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 101-136. | Zbl 0939.76063

[20] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-463. | Zbl 0697.65068

[21] B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source. Calcolo 38 (2001) 201-231. | Zbl 1008.65066

[22] P.A. Raviart, An analysis of particle methods, in Numerical methods in fluid dynamics (Como, 1983). Lect. Notes Math. 1127 (1985) 243-324. | Zbl 0598.76003

[23] A.J.C. De Saint-Venant, 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 03.0482.04

[24] P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21 (1984) 995-1011. | Zbl 0565.65048