Abstract
We prove in this paper the convergence of the Marker and cell (MAC) scheme for the discretization of the steady-state incompressible Navier-Stokes equations in primitive variables on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven; they yield the existence of discrete solutions and the compactness of sequences of solutions obtained with family of meshes the space step of which tends to zero. We then establish that the limit is a weak solution to the continuous problem.
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References
Blanc, P.: Error estimate for a finite volume scheme on a MAC mesh for the Stokes problem. In: Finite Volumes for Complex Applications II, pp. 117–124. Hermes Science Publishing, Paris (1999)
Chénier, E., Eymard R. nd Gallouët, T., Herbin, R.: An extension of the MAC scheme to locally refined meshes: convergence analysis for the full tensor time-dependent Navier-Stokes equations. Calcolo, to appear (2014)
Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Techniques of Scientific Computing, Part III, Handbook of Numerical Analysis, VII, pp. 713–1020. North-Holland, Amsterdam (2000)
Gallouët, T., Herbin, R., Latché, J.: \({W}^{1, q}\) stability of the Fortin operator for the MAC scheme. Calcolo 69, 63–71 (2012). See also http://hal.archives-ouvertes.fr/
Harlow, F., Welch, J.: Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface. Phys. Fluids 8, 2182–2189 (1965)
Herbin, R., Latché, J., Mallem, K.: Numerical analysis of the MAC scheme for the Navier-Stokes equations in primitive variables. (in preparation)
Nicolaïdes, R., Wu, X.: Analysis and convergence of the mac scheme ii, Navier-Stokes equations. Math. Comp. 65, 29–44 (1996)
Patankar, S.: Numerical heat transfer and fluid flow. Series in Computational Methods in Mechanics and Thermal Sciences, vol. XIII. Hemisphere Publishing Corporation, Washington (1980)
Wesseling, P.: Principles of Computational Fluid Dynamics. Springer, Berlin (2001)
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Herbin, R., Latché, JC., Mallem, K. (2014). Convergence of the MAC Scheme for the Steady-State Incompressible Navier-Stokes Equations on Non-uniform Grids. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds) Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects. Springer Proceedings in Mathematics & Statistics, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-319-05684-5_33
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DOI: https://doi.org/10.1007/978-3-319-05684-5_33
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