Summary
In this paper a priori error estimates are derived for the discretization error which results when the linear Navier-Stokes equations are solved by a method which closely resembles the MAC-method of Harlow and Welch. General boundary conditions are permitted and the estimates are in terms of the discreteL 2 norm. A solvability result is given which also applies to a generalization of the method to the nonlinear case. This generalization is used in the last section to produce a numerical solution to the problem of flow around an obstacle.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Batchelor, G.K.: An introduction to fluid dynamics. London: Cambridge University Press 1967
Berge, C., Ghouila-Houri, A.: Programming, games and transportation networks. London: Methuen 1965.
Chorin, A.J.: On the convergence of discrete approximations to the navier-stokes equations. Math. Comput.23, 341–353 (1969)
Daly, B.J.: Numerical study of the effects of surface tension on interface instability. Phys. Fluids12, 1340–1354 (1969)
Daly, B.J., Pracht, W.E.: Numerical study of density-current surges. Phys. Fluids11, 15–30 (1968)
Donovan, L.F.: A numerical solution of unsteady flow in a two-dimensional square cavity. AIAA J.8, 524–529 (1970)
Harlow, F.H., Welch, F.E.: Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids8, 2182–2189 (1965)
Hirt, C.W.: Heuristic stability theory for finite-difference equations. J. Computational Phys.2, 339–355 (1968)
Jamet, P., Lascaux, P., Raviart, P.A.: Une methode de resolution numerique des equations de navier-stokes. Numer. Math.16, 93–114 (1970)
Krzhivitski, A., Ladyzhenskaya, O.A.: A grid method for the navier-stokes equations. Soviet Physics Dokl.11, 212–213 (1966)
Leray, J.: Etudes de diverses equations integrales non lineaires et de quelques problems que pose l'hydrodynamique. J. Math. Pures Appl., 9 Serie,12, 1–82 (1933)
Prandtl, L., Tietjens, O.G.: Applied hydro- and aeromechanics. New York: McGraw-Hill 1934
Roache, P.J.: Computational fluid dynamics. Albuquerque: Hermosa 1976
Roache, P.J.: On artificial viscosity. J. Computational Phys.10, 169–187 (1972)
Richtmyer, R.D., Morton, K.W.: Difference methods for initial value problems, 2nd ed. New York: Interscience 1967
Temam, R.: Une methode d'approximation de la solution des equations de navier-stokes. Bull. Soc. Math. France96, 115–152 (1968)
Temam, R.: On the theory and numerical analysis of the navier-stokes equations. Univ. of Maryland, Lecture Note No. 9, 1973
Welch, J.E., Harlow, F.H., Shannon, J.P., Daly, B.J.: The MAC method. LASL Report No. LA-3425, Los Alamos Scientific Laboratory, Los Alamos, N. Mexico, 1966
Author information
Authors and Affiliations
Additional information
This work supported in part by Westinghouse Nuclear Energy Systems. Research Report #76-13
Rights and permissions
About this article
Cite this article
Porsching, T.A. Error estimates for MAC-like approximations to the linear Navier-Stokes equations. Numer. Math. 29, 291–306 (1978). https://doi.org/10.1007/BF01389214
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01389214