Hostname: page-component-6d856f89d9-8l2sj Total loading time: 0 Render date: 2024-07-16T06:27:01.948Z Has data issue: false hasContentIssue false
  • English
  • Français

A discrete kinetic approximation for the incompressible Navier-Stokes equations

Published online by Cambridge University Press:  12 January 2008

Maria Francesca Carfora  and
Roberto Natalini
Maria Francesca Carfora
Affiliation:
Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Via P. Castellino, 111, 80131, Napoli, Italia. f.carfora@iac.cnr.it
Roberto Natalini
Affiliation:
Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Viale del Policlinico 137, 00161 Roma, Italia. r.natalini@iac.cnr.it
Get access

Abstract

In this paper we introduce a new class of numerical schemes for the incompressible Navier-Stokes equations, which are inspired by the theory of discrete kinetic schemes for compressible fluids. For these approximations it is possible to give a stability condition, based on a discrete velocities version of the Boltzmann H-theorem. Numerical tests are performed to investigate their convergence and accuracy.

Keywords

Incompressible fluidskinetic schemesBGK modelsfinite difference schemes.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 1 , January 2008 , pp. 93 - 112
Copyright
© EDP Sciences, SMAI, 2008

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

Aregba-Driollet, D. and Natalini, R., Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37 (2000) 1973–2004. CrossRef
Aregba-Driollet, D., Natalini, R. and Tang, S., Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems. Math. Comp. 73 (2004) 63–94. CrossRef
Banda, M.K., Klar, A., Pareschi, L. and Seaid, M., Compressible and incompressible limits for hyperbolic systems with relaxation. J. Comput. Appl. Math. 168 (2004) 41–52. CrossRef
Bianchini, S., Hyperbolic limit of the Jin-Xin relaxation model. Comm. Pure Appl. Math. 59 (2006) 688–753. CrossRef
Brenier, Y., Natalini, R. and Puel, M., On a relaxation approximation of the incompressible Navier-Stokes equations. Proc. Amer. Math. Soc. 132 (2004) 1021–1028. CrossRef
Boghosian, B.M., Love, P.J., Coveney, P.V., Karlin, I.V., Succi, S. and Yepez, J., Galilean-invariant Lattice-Boltzmann models with H theorem. Phys. Rev. E 68 (2003) 25103–25106. CrossRef
Bouchut, F., Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Statist. Phys. 95 (1999) 113–170. CrossRef
Bouchut, F., Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. 94 (2003) 623–672. CrossRef
F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics series. Birkhäuser (2004).
Chorin, A.J., Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 745–762. CrossRef
Donatelli, D. and Marcati, P., Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems. Trans. Amer. Math. Soc. 356 (2004) 2093–2121. CrossRef
E, W. and Liu, J.G., Projection method. I. Convergence and numerical boundary layers. SIAM J. Numer. Anal. 32 (1995) 1017–1057; Projection method. II. Godunov-Ryabenki analysis. SIAM J. Numer. Anal. 33 (1996) 1597–1621. CrossRef
Hou, T.Y. and Wetton, B.T.R., Second-order convergence of a projection scheme for the incompressible Navier-Stokes equations with boundaries. SIAM J. Numer. Anal. 30 (1993) 609–629. CrossRef
Junk, M., Kinetic schemes in the case of low Mach numbers. J. Comput. Phys. 151 (1999) 947–968. CrossRef
Junk, M. and Klar, A., Discretization for the incompressible Navier-Stokes equations based on the Lattice Boltzmann method. SIAM J. Sci. Comp. 22 (2000) 1–19. CrossRef
Junk, M. and Yong, W.A., Rigorous Navier-Stokes limit of the Lattice Boltzmann equation. Asymptot. Anal. 35 (2003) 165–185.
Kim, J. and Moin, P., Application of a fractional-step method to incompressible Navier-Stokes. J. Comput. Phys. 59 (1985) 308–323. CrossRef
Natalini, R., A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws. J. Diff. Equation 148 (1998) 292–317. CrossRef
Natalini, R. and Rousset, F., Convergence of a singular Euler-Poisson approximation of the incompressible Navier-Stokes equations. Proc. Am. Math. Soc. 134 (2006) 2251–2258. CrossRef
B. Perthame, Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics and its Applications 21. Oxford University Press, Oxford (2002).
Reider, M. and Sterling, J., Accuracy of discrete velocity BGK models for the simulation of the incompressible Navier-Stokes equations. Comput. Fluids 24 (1995) 459–467. CrossRef
S. Succi, The Lattice Boltzmann Equation. Oxford University Press, Oxford (2001).
Temam, R., Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. I. Arch. Ration. Mech. Anal. 32 (1969) 135–153; Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II. Arch. Ration. Mech. Anal. 33 (1969) 377–385. CrossRef
Wetton, B.R., Analysis of the spatial error for a class of finite difference methods for viscous incompressible flow. SIAM J. Numer. Anal. 34 (1997) 723–755; Error analysis for Chorin's original fully discrete projection method and regularizations in space and time. SIAM J. Numer. Anal. 34 (1997) 1683–1697. CrossRef
D.A. Wolf-Gladrow, Lattice-gas cellular automata and Lattice Boltzmann models. An introduction, Lecture Notes in Mathematics 1725. Springer-Verlag, Berlin (2000).