Transport of Pollutant in Shallow Water A Two Time Steps Kinetic Method

Emmanuel Audusse; Marie-Odile Bristeau

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 2, page 389-416
  • ISSN: 0764-583X

Abstract

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The aim of this paper is to present a finite volume kinetic method to compute the transport of a passive pollutant by a flow modeled by the shallow water equations using a new time discretization that allows large time steps for the pollutant computation. For the hydrodynamic part the kinetic solver ensures – even in the case of a non flat bottom – the preservation of the steady state of a lake at rest, the non-negativity of the water height and the existence of an entropy inequality. On an other hand the transport computation ensures the conservation of pollutant mass, a non-negativity property and a maximum principle for the concentration of pollutant and the preservation of discrete steady states associated with the lake at rest equilibrium. The interest of the developed method is to preserve these theoretical properties with a scheme that allows to disconnect the hydrodynamic time step – related to a classical CFL condition – and the transport one – related to a new CFL condition – and further the hydrodynamic calculation and the transport one. The CPU time is very reduced and we can easily solve different transport problems with the same hydrodynamic solution without large storage. Moreover the numerical results exhibit a better accuracy than with a classical method especially when using 1D or 2D regular grids.

How to cite

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Audusse, Emmanuel, and Bristeau, Marie-Odile. "Transport of Pollutant in Shallow Water A Two Time Steps Kinetic Method." ESAIM: Mathematical Modelling and Numerical Analysis 37.2 (2010): 389-416. <http://eudml.org/doc/194170>.

@article{Audusse2010,
abstract = { The aim of this paper is to present a finite volume kinetic method to compute the transport of a passive pollutant by a flow modeled by the shallow water equations using a new time discretization that allows large time steps for the pollutant computation. For the hydrodynamic part the kinetic solver ensures – even in the case of a non flat bottom – the preservation of the steady state of a lake at rest, the non-negativity of the water height and the existence of an entropy inequality. On an other hand the transport computation ensures the conservation of pollutant mass, a non-negativity property and a maximum principle for the concentration of pollutant and the preservation of discrete steady states associated with the lake at rest equilibrium. The interest of the developed method is to preserve these theoretical properties with a scheme that allows to disconnect the hydrodynamic time step – related to a classical CFL condition – and the transport one – related to a new CFL condition – and further the hydrodynamic calculation and the transport one. The CPU time is very reduced and we can easily solve different transport problems with the same hydrodynamic solution without large storage. Moreover the numerical results exhibit a better accuracy than with a classical method especially when using 1D or 2D regular grids. },
author = {Audusse, Emmanuel, Bristeau, Marie-Odile},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Shallow water equations; Saint-Venant system; finite volume method; kinetic scheme; transport of pollutant; time discretization.; shallow water equations; time discretization},
language = {eng},
month = {3},
number = {2},
pages = {389-416},
publisher = {EDP Sciences},
title = {Transport of Pollutant in Shallow Water A Two Time Steps Kinetic Method},
url = {http://eudml.org/doc/194170},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Audusse, Emmanuel
AU - Bristeau, Marie-Odile
TI - Transport of Pollutant in Shallow Water A Two Time Steps Kinetic Method
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 2
SP - 389
EP - 416
AB - The aim of this paper is to present a finite volume kinetic method to compute the transport of a passive pollutant by a flow modeled by the shallow water equations using a new time discretization that allows large time steps for the pollutant computation. For the hydrodynamic part the kinetic solver ensures – even in the case of a non flat bottom – the preservation of the steady state of a lake at rest, the non-negativity of the water height and the existence of an entropy inequality. On an other hand the transport computation ensures the conservation of pollutant mass, a non-negativity property and a maximum principle for the concentration of pollutant and the preservation of discrete steady states associated with the lake at rest equilibrium. The interest of the developed method is to preserve these theoretical properties with a scheme that allows to disconnect the hydrodynamic time step – related to a classical CFL condition – and the transport one – related to a new CFL condition – and further the hydrodynamic calculation and the transport one. The CPU time is very reduced and we can easily solve different transport problems with the same hydrodynamic solution without large storage. Moreover the numerical results exhibit a better accuracy than with a classical method especially when using 1D or 2D regular grids.
LA - eng
KW - Shallow water equations; Saint-Venant system; finite volume method; kinetic scheme; transport of pollutant; time discretization.; shallow water equations; time discretization
UR - http://eudml.org/doc/194170
ER -

References

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  1. E. Audusse, M.O. Bristeau and B. Perthame, Kinetic schemes for Saint-Venant equations with source terms on unstructured grids. INRIA Report, RR-3989 (2000), http://www.inria.fr/RRRT/RR-3989.html.  
  2. A. Bermudez and M.E. Vasquez, Upwind methods for hyperbolic conservation laws with source terms. Comput. & Fluids23 (1994) 1049-1071.  
  3. M.O. Bristeau and B. Coussin, Boundary conditions for the shallow water equations solved by kinetic schemes. INRIA Report, RR-4282 (2001), http://www.inria.fr/RRRT/RR-4282.html.  
  4. M.O. Bristeau and B. Perthame, Transport of pollutant in shallow water using kinetic schemes. CEMRACS, ESAIM Proc. 10 (1999) 9-21, http://www.emath.fr/Maths/Proc/Vol.10.  
  5. R. Eymard, T. Gallouet and R. Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VIII, P.G. Ciarlet and J.L. Lions Eds., Amsterdam, North-Holland (2000).  
  6. T. Gallouet, J.M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow water equations with topography. Comput. & Fluids32 (2003) 479-513.  
  7. J.F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation. Discrete Contin. Dynam. Systems 1 (2001) 89-102.  
  8. E. Godlewski and P.A. Raviart, Numerical approximation of hyperbolic systems of conservation laws. Springer-Verlag, New York, Appl. Math. Sci. 118 (1996).  
  9. L. Gosse and A.Y. LeRoux, A well-balanced scheme designed for inhomogeneous scalar conservation laws. C. R. Acad. Sci. Paris Sér. I Math.323 (1996) 543-546.  
  10. J.M. Hervouet, Hydrodynamique des écoulements à surface libre, apport de la méthode des éléments finis. EDF (2001).  
  11. S. Jin, A steady state capturing method for hyperbolic system with geometrical source terms. ESAIM: M2AN35 (2001) 631-646.  
  12. R.J. LeVêque, Numerical Methods for Conservation Laws. Second edition, ETH Zurich, Birkhauser, Lectures in Mathematics (1992).  
  13. R.J. LeVêque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys.146 (1998) 346-365.  
  14. L. Martin, Fonctionnement écologique de la Seine à l'aval de la station d'épuration d'Achères: données expérimentales et modélisation bidimensionnelle. Ph.D. Thesis, École des Mines de Paris, France (2001).  
  15. B. Perthame, Kinetic formulations of conservation laws. Oxford University Press (2002).  
  16. B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term. Calcolo38 (2001) 201-231.  
  17. P.L. Roe, Upwind differencing schemes for hyperbolic conservation laws with source terms, in Nonlinear Hyperbolic Problems, C. Carasso, P.A. Raviart and D. Serre Eds., Berlin, Springer-Verlag, Lecture Notes in Math. 1270 (1987) 41-51.  
  18. A.J.C. de Saint-Venant, Théorie du mouvement non permanent des eaux, avec application aux crues de rivières et à l'introduction des marées dans leur lit. C. R. Acad. Sci. Paris Sér. I Math.73 (1871) 147-154.  
  19. J.J. Stoker, The formation of breakers and bores. Comput. Appl. Math. 1 (1948).  

Citations in EuDML Documents

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  1. Olivier Bernard, Anne-Céline Boulanger, Marie-Odile Bristeau, Jacques Sainte-Marie, A 2D model for hydrodynamics and biology coupling applied to algae growth simulations
  2. Alina Chertock, Alexander Kurganov, On a hybrid finite-volume-particle method
  3. Alina Chertock, Alexander Kurganov, On a hybrid finite-volume-particle method
  4. Emmanuel Audusse, Marie-Odile Bristeau, Benoît Perthame, Jacques Sainte-Marie, A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation
  5. Emmanuel Audusse, Marie-Odile Bristeau, Benoît Perthame, Jacques Sainte-Marie, A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation

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