On the error estimates of the vector penalty-projection methods: Second-order scheme
HTML articles powered by AMS MathViewer
- by Philippe Angot and Rima Cheaytou PDF
- Math. Comp. 87 (2018), 2159-2187 Request permission
Abstract:
In this paper, we study the vector penalty-projection method for incompressible unsteady Stokes equations with Dirichlet boundary conditions. The time derivative is approximated by the backward difference formula of second-order scheme (BDF2), namely Gear’s scheme, whereas the approximation in space is performed by the finite volume scheme on a Marker And Cell (MAC) grid. After proving the stability of the method, we show that it yields second-error estimates in the time step for both velocity and pressure in the norm of $l^{\infty }(\mathbf {L}^2(\Omega ))$ and $l^2(L^2(\Omega ))$, respectively. Also, we show that the splitting error for both velocity and pressure is of order $\mathcal {O}(\sqrt {\varepsilon \delta t})$ where $\varepsilon$ is a penalty parameter chosen as small as desired and $\delta t$ is the time step. Numerical results in agreement with the theoretical study are also provided.References
- P. Angot, J.-P. Caltagirone, and P. Fabrie, Analysis for the fast vector penalty-projection solver of incompressible multiphase Navier-Stokes/brinkman problems, Numer. Math.
- P. Angot, J.-P. Caltagirone, and P. Fabrie, A kinematic vector penalty-projection method for incompressible flow with variable density, C. R. Math. Acad. Sci. Paris, Ser. I.
- Philippe Angot, Jean-Paul Caltagirone, and Pierre Fabrie, Vector penalty-projection methods for the solution of unsteady incompressible flows, Finite volumes for complex applications V, ISTE, London, 2008, pp. 169–176. MR 2451404
- Philippe Angot, Jean-Paul Caltagirone, and Pierre Fabrie, A spectacular vector penalty-projection method for Darcy and Navier-Stokes problems, Finite volumes for complex applications VI. Problems & perspectives. Volume 1, 2, Springer Proc. Math., vol. 4, Springer, Heidelberg, 2011, pp. 39–47. MR 2815627, DOI 10.1007/978-3-642-20671-9_{5}
- Philippe Angot, Jean-Paul Caltagirone, and Pierre Fabrie, A fast vector penalty-projection method for incompressible non-homogeneous or multiphase Navier-Stokes problems, Appl. Math. Lett. 25 (2012), no. 11, 1681–1688. MR 2957735, DOI 10.1016/j.aml.2012.01.037
- Philippe Angot, Jean-Paul Caltagirone, and Pierre Fabrie, A new fast method to compute saddle-points in constrained optimization and applications, Appl. Math. Lett. 25 (2012), no. 3, 245–251. MR 2855967, DOI 10.1016/j.aml.2011.08.015
- Philippe Angot, Jean-Paul Caltagirone, and Pierre Fabrie, Fast discrete Helmholtz-Hodge decompositions in bounded domains, Appl. Math. Lett. 26 (2013), no. 4, 445–451. MR 3019973, DOI 10.1016/j.aml.2012.11.006
- P. Angot and R. Cheaytou, Vector penalty-projection methods for incompressible fluid flows with open boundary conditions, in Algoritmy 2012 - (A. Handlovicočá et al. Eds), Slovak University of Technology in Bratislava, Publishing House of STU (Bratislava) (2012), 219–229.
- P. Angot and R. Cheaytou, Vector penalty-projection methods for outflow boundary conditions with optimal second-order accuracy (under revision), Comput. Fluids (2016).
- Ph. Angot, M. Jobelin, and J.-C. Latché, Error analysis of the penalty-projection method for the time dependent Stokes equations, Int. J. Finite Vol. 6 (2009), no. 1, 26. MR 2471404
- Franck Boyer and Pierre Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, vol. 183, Springer, New York, 2013. MR 2986590, DOI 10.1007/978-1-4614-5975-0
- 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
- J.-P. Caltagirone and J. Breil, Sur une méthode de projection vectorielle pour la résolution des équations de Navier-Stokes, C. R. Math. Acad. Sci. Paris 327 (1999), no. 11, 1179–1184.
- R. Cheaytou, Etude des méthodes de pénalité-projection vectorielle pour les équations de navier-stokes avec conditions aux limites ouvertes,, Ph.D. thesis, Université d’Aix-Marseille, April 2014.
- 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
- 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
- J. H. Ferziger and M. Perić, Computational methods for fluid dynamics, Springer-Verlag, Berlin, 1996. MR 1384758, DOI 10.1007/978-3-642-97651-3
- C. Févrière, J. Laminie, P. Poullet, and Ph. Angot, On the penalty-projection method for the Navier-Stokes equations with the MAC mesh, J. Comput. Appl. Math. 226 (2009), no. 2, 228–245. MR 2501639, DOI 10.1016/j.cam.2008.08.014
- Michel Fortin and Roland Glowinski, Augmented Lagrangian methods, Studies in Mathematics and its Applications, vol. 15, North-Holland Publishing Co., Amsterdam, 1983. Applications to the numerical solution of boundary value problems; Translated from the French by B. Hunt and D. C. Spicer. MR 724072
- K. Goda, A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows, J. Comput. Phys. 30 (1979), no. 1, 76–95.
- 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, P. Minev, and J. Shen, Error analysis of pressure-correction schemes for the time-dependent Stokes equations with open boundary conditions, SIAM J. Numer. Anal. 43 (2005), no. 1, 239–258. MR 2177143, DOI 10.1137/040604418
- Jean-Luc Guermond and Jie Shen, Quelques résultats nouveaux sur les méthodes de projection, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 12, 1111–1116 (French, with English and French summaries). MR 1881243, DOI 10.1016/S0764-4442(01)02157-7
- J. L. Guermond and Jie Shen, A new class of truly consistent splitting schemes for incompressible flows, J. Comput. Phys. 192 (2003), no. 1, 262–276. MR 2045709, DOI 10.1016/j.jcp.2003.07.009
- J. L. Guermond and Jie Shen, Velocity-correction projection methods for incompressible flows, SIAM J. Numer. Anal. 41 (2003), no. 1, 112–134. MR 1974494, DOI 10.1137/S0036142901395400
- J. L. Guermond and Jie Shen, On the error estimates for the rotational pressure-correction projection methods, Math. Comp. 73 (2004), no. 248, 1719–1737. MR 2059733, DOI 10.1090/S0025-5718-03-01621-1
- J. L. Guermond, P. Minev, and Jie Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 44-47, 6011–6045. MR 2250931, DOI 10.1016/j.cma.2005.10.010
- J. L. Guermond, Jie Shen, and Xiaofeng Yang, Error analysis of fully discrete velocity-correction methods for incompressible flows, Math. Comp. 77 (2008), no. 263, 1387–1405. MR 2398773, DOI 10.1090/S0025-5718-08-02109-1
- Francis H. Harlow and J. Eddie Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids 8 (1965), no. 12, 2182–2189. MR 3155392, DOI 10.1063/1.1761178
- M. Jobelin, C. Lapuerta, J.-C. Latché, Ph. Angot, and B. Piar, A finite element penalty-projection method for incompressible flows, J. Comput. Phys. 217 (2006), no. 2, 502–518. MR 2260612, DOI 10.1016/j.jcp.2006.01.019
- Hans Johnston and Jian-Guo Liu, Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term, J. Comput. Phys. 199 (2004), no. 1, 221–259. MR 2081004, DOI 10.1016/j.jcp.2004.02.009
- 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
- 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
- O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Mathematics and its Applications, Vol. 2, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Second English edition, revised and enlarged; Translated from the Russian by Richard A. Silverman and John Chu. MR 0254401
- J. Leray, Essai sur les mouvements plans d’un liquide visqueux que limitent des parois, J. Math. Pures Appl. 13 (1934), 331–418.
- S.A. Orszag, M. Israeli, and M.O. Deville, Boundary conditions for incompressible flows, J. Sci. Comput. 1 (1986), 75–111.
- A. Poux, S. Glockner, E. Ahusborde, and M. Azaïez, Open boundary conditions for the velocity-correction scheme of the Navier-Stokes equations, Comput. & Fluids 70 (2012), 29–43. MR 2988624, DOI 10.1016/j.compfluid.2012.08.028
- Jae-Hong Pyo and Jie Shen, Normal mode analysis of second-order projection methods for incompressible flows, Discrete Contin. Dyn. Syst. Ser. B 5 (2005), no. 3, 817–840. MR 2151734, DOI 10.3934/dcdsb.2005.5.817
- 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 some higher order projection and penalty-projection methods for Navier-Stokes equations, Numer. Math. 62 (1992), no. 1, 49–73. MR 1159045, DOI 10.1007/BF01396220
- 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
- Jie Shen and Xiaofeng Yang, Error estimates for finite element approximations of consistent splitting schemes for incompressible flows, Discrete Contin. Dyn. Syst. Ser. B 8 (2007), no. 3, 663–676. MR 2328729, DOI 10.3934/dcdsb.2007.8.663
- 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, Revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979. Theory and numerical analysis; With an appendix by F. Thomasset. MR 603444
- L. J. P. Timmermans, P. D. Minev, and F. N. van de Vosse, An approximate projection scheme for incompressible flow using spectral elements, Internat. J. Numer. Methods Fluids 22 (1996), no. 7, 673–688. MR 3363448, DOI 10.1002/(SICI)1097-0363(19960415)22:7<673::AID-FLD373>3.0.CO;2-O
Additional Information
- Philippe Angot
- Affiliation: Aix-Marseille Université, Institut de Mathématiques de Marseille (I2M) - CNRS UMR7373, Centrale Marseille, 13453 Marseille cedex 13 - France
- MR Author ID: 328099
- Email: philippe.angot@univ-amu.fr
- Rima Cheaytou
- Affiliation: Aix-Marseille Université, Institut de Mathématiques de Marseille (I2M) - CNRS UMR7373, Centrale Marseille, 13453 Marseille cedex 13 - France
- Email: rima.cheaytou@gmail.com
- Received by editor(s): November 18, 2016
- Received by editor(s) in revised form: April 12, 2017
- Published electronically: December 27, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 87 (2018), 2159-2187
- MSC (2010): Primary 76D07, 35Q30, 65M15, 65M12
- DOI: https://doi.org/10.1090/mcom/3309
- MathSciNet review: 3802431