On the error estimates of the vector penalty-projection methods: Second-order scheme

Angot, Philippe; Cheaytou, Rima

doi:10.1090/mcom/3309

On the error estimates of the vector penalty-projection methods: Second-order scheme
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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