×

A particle method to solve the Navier-Stokes system. (English) Zbl 0707.76029

Summary: We extend to the case of the two-dimensional Navier-Stokes equations, a particle method introduced in a previous paper [the authors, C. R. Acad. Sci., Paris, Ser. I 297, 133-136 (1983; Zbl 0534.65077)] to solve linear convection-diffusion equations. The method is based on a viscous splitting of the operator. The particles move under the effect of the velocity field but are not affected by the diffusion which is taken into account by the weights. We prove the stability and the convergence of the method.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
76M25 Other numerical methods (fluid mechanics) (MSC2010)

Citations:

Zbl 0534.65077
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Beale, J.T., Majda, A.: Vortex methods I: Convergence in three dimensions. Math. Comput.39, 1-27 (1982) · Zbl 0488.76024
[2] Beale, J.T., Majda, A.: Vortex methods II: Higher order accuracy in two and three dimensions. Math. Comput.39, 29-52 (1982) · Zbl 0488.76025
[3] Beale, J.T., Majda, A.: Rates of convergence for viscous splitting of the Navier-Stokes equations. Math. Comput.37, 243-259 (1981) · Zbl 0518.76027
[4] Bers, L., John, F., Schechter, M.: Partial differential equations. American Mathematical Society, Providence (1964) · Zbl 0126.00207
[5] Choquin, J.P., Huberson, S.: Particle simulations of viscous flows for Navier-Stokes equations. Comput. Fluids (to appear) · Zbl 0596.76045
[6] Cottet, G.H.: A new approach for the analysis of vortex methods in two and three dimensions. Ann. Inst. Henri Poincaré, Anal. Non Linéaire5, 227-285 (1988) · Zbl 0688.76017
[7] Cottet, G.H., Gallic, S.: A particle method to solve transport-diffusion equations I: the linear case. Internal Report 115, C.M.A.P., Ecole Polytechnique, Palaiseau, France and C.R. Acad. Sci., Paris, Sér. I297, 133-136 (1983) · Zbl 0534.65077
[8] Goodman, J.: Convergence of the random vortex methods. Commun. Pure Appl. Math.40, 189-220 (1987) · Zbl 0635.35077
[9] Long, D.G.: Convergence of random vortex methods. Ph.D. Thesis, Berkeley (1986)
[10] Lucquin-Desreux, B.: Particle approximation of the two dimensional Euler and Navier-Stokes equations. Rech. Aérosp.4, 1-12 (1987) · Zbl 0619.76031
[11] McGrath, F.J.: Nonstationary plane flow of viscous and ideal fluids. Arch. Rat. Mech. Anal.27, 328-348 (1986) · Zbl 0187.49508
[12] Mas-Gallic, S., Raviart, P.A.: Particle approximation of convection-diffusion problems. Internal Report R86013, lab. Anal. Num., Université Pierre et Marie Curie, Paris, France (1986) and C. R. Acad. Sci., Paris, Sér. I305, 431-434 (1987)
[13] Raviart, P.A.: An analysis of particle methods. In: (Brezzi, F. (ed)) Numerical Methods in Fluid Dynamics. Lecture Notes in Mathematics, vol. 1127. Berlin-Heidelberg-New York: Springer 1985 · Zbl 0598.76003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.