On a hybrid finite-volume-particle method

Alina Chertock; Alexander Kurganov

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 6, page 1071-1091
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Chertock, Alina, and Kurganov, Alexander. "On a hybrid finite-volume-particle method." ESAIM: Mathematical Modelling and Numerical Analysis 38.6 (2010): 1071-1091. <http://eudml.org/doc/194248>.

@article{Chertock2010,
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 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. },
author = {Chertock, Alina, Kurganov, Alexander},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Shallow water equations; transport of passive pollutant; finite-volume schemes; particle method.; numerical examples; transport of pollutant},
language = {eng},
month = {3},
number = {6},
pages = {1071-1091},
publisher = {EDP Sciences},
title = {On a hybrid finite-volume-particle method},
url = {http://eudml.org/doc/194248},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Chertock, Alina
AU - Kurganov, Alexander
TI - On a hybrid finite-volume-particle method
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 6
SP - 1071
EP - 1091
AB - 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.
LA - eng
KW - Shallow water equations; transport of passive pollutant; finite-volume schemes; particle method.; numerical examples; transport of pollutant
UR - http://eudml.org/doc/194248
ER -

References

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  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.  Zbl1133.65308
  2. E. Audusse and M.-O. Bristeau, Transport of pollutant in shallow water. A two time steps kinetic method. ESAIM: M2AN37 (2003) 389–416.  Zbl1137.65392
  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.  Zbl1034.65068
  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.  Zbl1151.76593
  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).  Zbl1101.76036
  6. A. Cohen and B. Perthame, Optimal approximations of transport equations by particle and pseudoparticle methods. SIAM J. Math. Anal.32 (2000) 616–636.  Zbl0972.65058
  7. B. Engquist, P. Lötstedt and B. Sjögreen, Nonlinear filters for efficient shock computation. Math. Comp.52 (1989) 509–537.  
  8. A.F. Filippov, Differential equations with discontinuous right-hand side. (Russian). Mat. Sb. (N.S.)51 (1960) 99–128.  Zbl0138.32204
  9. A.F. Filippov, Differential equations with discontinuous right-hand side. AMS Transl.42 (1964) 199–231.  Zbl0148.33002
  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).  Zbl0664.34001
  11. T. Gallouët, J.-M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids32 (2003) 479–513.  
  12. J.F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B1 (2001) 89–102.  Zbl0997.76023
  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.  Zbl0967.65098
  14. A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN36 (2002) 397–425.  Zbl1137.65398
  15. A. Kurganov and C.-T. Lin, On the reduction of numerical dissipation in central-upwind schemes (in preparation).  Zbl1164.65455
  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.  Zbl0998.65091
  17. A. Kurganov and G. Petrova, Central schemes and contact discontinuities. ESAIM: M2AN34 (2000) 1259–1275.  Zbl0972.65055
  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.  Zbl0987.65085
  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.  
  20. H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys.87 (1990) 408–463.  Zbl0697.65068
  21. B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source. Calcolo38 (2001) 201–231.  Zbl1008.65066
  22. P.A. Raviart, An analysis of particle methods, in Numerical methods in fluid dynamics (Como, 1983). Lect. Notes Math.1127 (1985) 243–324.  
  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. Paris73 (1871) 147–154.  Zbl03.0482.04
  24. P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal.21 (1984) 995–1011.  Zbl0565.65048

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