Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T11:16:30.508Z Has data issue: false hasContentIssue false
  • English
  • Français

On a hybrid finite-volume-particle method

Published online by Cambridge University Press:  15 December 2004

Alina Chertock  and
Alexander Kurganov
Alina Chertock
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA. [email protected].
Alexander Kurganov
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA. [email protected].
Get access

Abstract

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 utilizedat 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 extendthe finite-volume-particle method to the two-dimensional case.

Keywords

Shallow water equationstransport of passive pollutantfinite-volume schemesparticle method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 6 , November 2004 , pp. 1071 - 1091
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R. and Perthame, B., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050–2065. CrossRef
Audusse, E. and Bristeau, M.-O., Transport of pollutant in shallow water. A two time steps kinetic method. ESAIM: M2AN 37 (2003) 389–416. CrossRef
Bale, D.S., LeVeque, R.J., Mitran, S. and Rossmanith, J.A., A wave propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 24 (2002) 955–978. CrossRef
Bristeau, M.-O. and Perthame, B., Transport of pollutant in shallow water using kinetic schemes. CEMRACS, Orsay (electronic), ESAIM Proc., Paris. Soc. Math. Appl. Indust. 10 (1999) 9–21.
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).
Cohen, A. and Perthame, B., Optimal approximations of transport equations by particle and pseudoparticle methods. SIAM J. Math. Anal. 32 (2000) 616–636. CrossRef
Engquist, B., Lötstedt, P. and Sjögreen, B., Nonlinear filters for efficient shock computation. Math. Comp. 52 (1989) 509–537. CrossRef
Filippov, A.F., Differential equations with discontinuous right-hand side. (Russian). Mat. Sb. (N.S.) 51 (1960) 99–128.
Filippov, A.F., Differential equations with discontinuous right-hand side. AMS Transl. 42 (1964) 199–231.
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).
Gallouët, T., Hérard, J.-M. and Seguin, N., Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32 (2003) 479–513. CrossRef
Gerbeau, J.F. and Perthame, B., Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 89–102.
Gottlieb, S., Shu, C.-W. and Tadmor, E., High order time discretization methods with the strong stability property. SIAM Rev. 43 (2001) 89–112. CrossRef
Kurganov, A. and Levy, D., Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397–425. CrossRef
A. Kurganov and C.-T. Lin, On the reduction of numerical dissipation in central-upwind schemes (in preparation).
Kurganov, A., Noelle, S. and Petrova, G., Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21 (2001) 707–740. CrossRef
Kurganov, A. and Petrova, G., Central schemes and contact discontinuities. ESAIM: M2AN 34 (2000) 1259–1275. CrossRef
Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241–282. CrossRef
van Leer, B., Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 101–136. CrossRef
Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408–463. CrossRef
Perthame, B. and Simeoni, C., A kinetic scheme for the Saint-Venant system with a source. Calcolo 38 (2001) 201–231. CrossRef
Raviart, P.A., An analysis of particle methods, in Numerical methods in fluid dynamics (Como, 1983). Lect. Notes Math. 1127 (1985) 243–324. CrossRef
de Saint-Venant, A.J.C., 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.
Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21 (1984) 995–1011. CrossRef