On the error estimates for the rotational pressure-correction projection methods
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- by J. L. Guermond and Jie Shen;
- Math. Comp. 73 (2004), 1719-1737
- DOI: https://doi.org/10.1090/S0025-5718-03-01621-1
- Published electronically: December 19, 2003
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Abstract:
In this paper we study the rotational form of the pressure-correction method that was proposed by Timmermans, Minev, and Van De Vosse. We show that the rotational form of the algorithm provides better accuracy in terms of the $H^1$-norm of the velocity and of the $L^2$-norm of the pressure than the standard form.References
- David L. Brown, Ricardo Cortez, and Michael L. Minion, Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys. 168 (2001), no. 2, 464–499. MR 1826523, DOI 10.1006/jcph.2001.6715
- Alexandre Joel Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp. 22 (1968), 745–762. MR 242392, DOI 10.1090/S0025-5718-1968-0242392-2
- Alexandre Joel Chorin, On the convergence of discrete approximations to the Navier-Stokes equations, Math. Comp. 23 (1969), 341–353. MR 242393, DOI 10.1090/S0025-5718-1969-0242393-5
- Weinan E and Jian-Guo Liu, Projection method. I. Convergence and numerical boundary layers, SIAM J. Numer. Anal. 32 (1995), no. 4, 1017–1057. MR 1342281, DOI 10.1137/0732047
- Weinan E and Jian-Guo Liu, Projection method. III. Spatial discretization on the staggered grid, Math. Comp. 71 (2002), no. 237, 27–47. MR 1862987, DOI 10.1090/S0025-5718-01-01313-8
- K. Goda. A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows. J. Comput. Phys., 30:76–95, 1979.
- Jean-Luc Guermond, Un résultat de convergence d’ordre deux en temps pour l’approximation des équations de Navier-Stokes par une technique de projection incrémentale, M2AN Math. Model. Numer. Anal. 33 (1999), no. 1, 169–189 (French, with English and French summaries). MR 1685751, DOI 10.1051/m2an:1999101
- J.L. Guermond and Jie Shen. Velocity-correction projection methods for incompressible flows. To appear in SIAM J. Numer. Anal.
- George Em. Karniadakis, Moshe Israeli, and Steven A. Orszag, High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys. 97 (1991), no. 2, 414–443. MR 1137607, DOI 10.1016/0021-9991(91)90007-8
- S. A. Orszag, M. Israeli, and M. Deville. Boundary conditions for incompressible flows. J. Sci. Comput., 1:75–111, 1986.
- Rolf Rannacher, On Chorin’s projection method for the incompressible Navier-Stokes equations, The Navier-Stokes equations II—theory and numerical methods (Oberwolfach, 1991) Lecture Notes in Math., vol. 1530, Springer, Berlin, 1992, pp. 167–183. MR 1226515, DOI 10.1007/BFb0090341
- Jie Shen, On error estimates of projection methods for Navier-Stokes equations: first-order schemes, SIAM J. Numer. Anal. 29 (1992), no. 1, 57–77. MR 1149084, DOI 10.1137/0729004
- Jie Shen. On pressure stabilization method and projection method for unsteady Navier-Stokes equations. In R. Vichnevetsky, D. Knight, and G. Richter, editors, Advances in Computer Methods for Partial Differential Equations, pages 658–662, IMACS, 1992.
- Jie Shen, Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput. 15 (1994), no. 6, 1489–1505. MR 1298626, DOI 10.1137/0915089
- Jie Shen, On error estimates of the projection methods for the Navier-Stokes equations: second-order schemes, Math. Comp. 65 (1996), no. 215, 1039–1065. MR 1348047, DOI 10.1090/S0025-5718-96-00750-8
- John C. Strikwerda and Young S. Lee, The accuracy of the fractional step method, SIAM J. Numer. Anal. 37 (1999), no. 1, 37–47. MR 1721264, DOI 10.1137/S0036142997326938
- R. Témam, Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II, Arch. Rational Mech. Anal. 33 (1969), 377–385 (French). MR 244654, DOI 10.1007/BF00247696
- Roger Temam, Navier-Stokes equations. Theory and numerical analysis, Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. MR 609732
- L. J. P. Timmermans, P. D. Minev, and F. N. Van De Vosse. An approximate projection scheme for incompressible flow using spectral elements. Int. J. Numer. Methods Fluids, 22:673–688, 1996.
- J. van Kan, A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J. Sci. Statist. Comput. 7 (1986), no. 3, 870–891. MR 848569, DOI 10.1137/0907059
Bibliographic Information
- J. L. Guermond
- Affiliation: LIMSI (CNRS-UPR 3152), BP 133, 91403, Orsay, France
- Email: guermond@limsi.fr
- Jie Shen
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 257933
- ORCID: 0000-0002-4885-5732
- Email: shen@math.purdue.edu
- Received by editor(s): February 11, 2002
- Received by editor(s) in revised form: March 2, 2003
- Published electronically: December 19, 2003
- Additional Notes: The work of the second author is partially supported by NFS grants DMS-0074283 and DMS-0311915. Part of the work was completed while this author was a CNRS “Poste Rouge” visitor at LIMSI
- © Copyright 2003 American Mathematical Society
- Journal: Math. Comp. 73 (2004), 1719-1737
- MSC (2000): Primary 65M12, 35Q30, 76D05
- DOI: https://doi.org/10.1090/S0025-5718-03-01621-1
- MathSciNet review: 2059733