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 fluids, kinetic schemes, BGK models, finite difference schemes
@article{M2AN_2008__42_1_93_0,
author = {Carfora, Maria Francesca and Natalini, Roberto},
title = {A discrete kinetic approximation for the incompressible {Navier-Stokes} equations},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {93--112},
year = {2008},
publisher = {EDP Sciences},
volume = {42},
number = {1},
doi = {10.1051/m2an:2007055},
mrnumber = {2387423},
zbl = {1135.76037},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an:2007055/}
}
TY - JOUR AU - Carfora, Maria Francesca AU - Natalini, Roberto TI - A discrete kinetic approximation for the incompressible Navier-Stokes equations JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 93 EP - 112 VL - 42 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an:2007055/ DO - 10.1051/m2an:2007055 LA - en ID - M2AN_2008__42_1_93_0 ER -
%0 Journal Article %A Carfora, Maria Francesca %A Natalini, Roberto %T A discrete kinetic approximation for the incompressible Navier-Stokes equations %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 93-112 %V 42 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an:2007055/ %R 10.1051/m2an:2007055 %G en %F M2AN_2008__42_1_93_0
Carfora, Maria Francesca; Natalini, Roberto. A discrete kinetic approximation for the incompressible Navier-Stokes equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 1, pp. 93-112. doi: 10.1051/m2an:2007055
[1] and , Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37 (2000) 1973-2004. | Zbl | MR
[2] , and , Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems. Math. Comp. 73 (2004) 63-94. | Zbl | MR
[3] , , and , Compressible and incompressible limits for hyperbolic systems with relaxation. J. Comput. Appl. Math. 168 (2004) 41-52. | Zbl | MR
[4] , Hyperbolic limit of the Jin-Xin relaxation model. Comm. Pure Appl. Math. 59 (2006) 688-753. | Zbl | MR
[5] , and , On a relaxation approximation of the incompressible Navier-Stokes equations. Proc. Amer. Math. Soc. 132 (2004) 1021-1028. | Zbl | MR
[6] , , , , and , Galilean-invariant Lattice-Boltzmann models with H theorem. Phys. Rev. E 68 (2003) 25103-25106.
[7] , Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Statist. Phys. 95 (1999) 113-170. | Zbl | MR
[8] , Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. 94 (2003) 623-672. | Zbl | MR
[9] , Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics series. Birkhäuser (2004). | Zbl | MR
[10] , Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 745-762. | Zbl | MR
[11] and , Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems. Trans. Amer. Math. Soc. 356 (2004) 2093-2121. | Zbl | MR
[12] W. E and J.G. Liu, 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. | Zbl | MR
[13] and , Second-order convergence of a projection scheme for the incompressible Navier-Stokes equations with boundaries. SIAM J. Numer. Anal. 30 (1993) 609-629. | Zbl | MR
[14] , Kinetic schemes in the case of low Mach numbers. J. Comput. Phys. 151 (1999) 947-968. | Zbl | MR
[15] and , Discretization for the incompressible Navier-Stokes equations based on the Lattice Boltzmann method. SIAM J. Sci. Comp. 22 (2000) 1-19. | Zbl | MR
[16] and , Rigorous Navier-Stokes limit of the Lattice Boltzmann equation. Asymptot. Anal. 35 (2003) 165-185. | Zbl | MR
[17] and , Application of a fractional-step method to incompressible Navier-Stokes. J. Comput. Phys. 59 (1985) 308-323. | Zbl | MR
[18] , A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws. J. Diff. Equation 148 (1998) 292-317. | Zbl | MR
[19] and , Convergence of a singular Euler-Poisson approximation of the incompressible Navier-Stokes equations. Proc. Am. Math. Soc. 134 (2006) 2251-2258. | MR
[20] , Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics and its Applications 21. Oxford University Press, Oxford (2002). | Zbl | MR
[21] and , Accuracy of discrete velocity BGK models for the simulation of the incompressible Navier-Stokes equations. Comput. Fluids 24 (1995) 459-467. | Zbl | MR
[22] , The Lattice Boltzmann Equation. Oxford University Press, Oxford (2001). | MR
[23] , 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. | Zbl | MR
[24] , 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. | Zbl | MR
[25] , Lattice-gas cellular automata and Lattice Boltzmann models. An introduction, Lecture Notes in Mathematics 1725. Springer-Verlag, Berlin (2000). | Zbl | MR
Cité par Sources :
