Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 
 
 

 

A convergent staggered scheme for the variable density incompressible Navier-Stokes equations


Authors: J. C. Latché and K. Saleh
Journal: Math. Comp. 87 (2018), 581-632
MSC (2010): Primary 35Q30, 35Q55, 65N12, 76M10, 76M12
DOI: https://doi.org/10.1090/mcom/3241
Published electronically: August 7, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we analyze a scheme for the time-dependent variable density Navier-Stokes equations. The algorithm is implicit in time, and the space approximation is based on a low-order staggered non-conforming finite element, the so-called Rannacher-Turek element. The convection term in the momentum balance equation is discretized by a finite volume technique, in such a way that a solution obeys a discrete kinetic energy balance, and the mass balance is approximated by an upwind finite volume method. We first show that the scheme preserves the stability properties of the continuous problem ( $ \mathrm {L}^\infty $-estimate for the density, $ \mathrm {L}^\infty (\mathrm {L}^2)$- and $ \mathrm {L}^2(\mathrm {H}^1)$-estimates for the velocity), which yields, by a topological degree technique, the existence of a solution. Then, invoking compactness arguments and passing to the limit in the scheme, we prove that any sequence of solutions (obtained with a sequence of discretizations the space and time step of which tend to zero) converges up to the extraction of a subsequence to a weak solution of the continuous problem.


References [Enhancements On Off] (What's this?)

  • [1] G. Ansanay-Alex, F. Babik, J. C. Latché, and D. Vola, An $ L^2$-stable approximation of the Navier-Stokes convection operator for low-order non-conforming finite elements, Internat. J. Numer. Methods Fluids 66 (2011), no. 5, 555-580. MR 2839213, https://doi.org/10.1002/fld.2270
  • [2] Fabrice Babik, Jean-Claude Latché, Bruno Piar, and Khaled Saleh, A staggered scheme with non-conforming refinement for the Navier-Stokes equations, Finite volumes for complex applications. VII. Methods and theoretical aspects, Springer Proc. Math. Stat., vol. 77, Springer, Cham, 2014, pp. 87-95. MR 3213339, https://doi.org/10.1007/978-3-319-05684-5_7
  • [3] F. Boyer, F. Dardalhon, C. Lapuerta, and J.-C. Latché, Stability of a Crank-Nicolson pressure correction scheme based on staggered discretizations, Internat. J. Numer. Methods Fluids 74 (2014), no. 1, 34-58. MR 3146663, https://doi.org/10.1002/fld.3837
  • [4] 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
  • [5] Susanne C. Brenner, Korn's inequalities for piecewise $ H^1$ vector fields, Math. Comp. 73 (2004), no. 247, 1067-1087. MR 2047078, https://doi.org/10.1090/S0025-5718-03-01579-5
  • [6] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33-75. MR 0343661
  • [7] Klaus Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. MR 787404
  • [8] Olivier Desjardins, Guillaume Blanquart, Guillaume Balarac, and Heinz Pitsch, High order conservative finite difference scheme for variable density low Mach number turbulent flows, J. Comput. Phys. 227 (2008), no. 15, 7125-7159. MR 2433965, https://doi.org/10.1016/j.jcp.2008.03.027
  • [9] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511-547. MR 1022305, https://doi.org/10.1007/BF01393835
  • [10] R. Eymard, T. Gallouët, M. Ghilani, and R. Herbin, Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes, IMA J. Numer. Anal. 18 (1998), no. 4, 563-594. MR 1681074, https://doi.org/10.1093/imanum/18.4.563
  • [11] Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713-1020. MR 1804748
  • [12] R. Eymard, T. Gallouët, and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal. 30 (2010), no. 4, 1009-1043. MR 2727814, https://doi.org/10.1093/imanum/drn084
  • [13] Thierry Gallouët, Laura Gastaldo, Raphaele Herbin, and Jean-Claude Latché, An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations, M2AN Math. Model. Numer. Anal. 42 (2008), no. 2, 303-331. MR 2405150, https://doi.org/10.1051/m2an:2008005
  • [14] T. Gallouët and J.-C. Latché, Compactness of discrete approximate solutions to parabolic PDEs--application to a turbulence model, Commun. Pure Appl. Anal. 11 (2012), no. 6, 2371-2391. MR 2912752, https://doi.org/10.3934/cpaa.2012.11.2371
  • [15] Laura Gastaldo, Raphaèle Herbin, and Jean-Claude Latché, An unconditionally stable finite element-finite volume pressure correction scheme for the drift-flux model, M2AN Math. Model. Numer. Anal. 44 (2010), no. 2, 251-287. MR 2655950, https://doi.org/10.1051/m2an/2010002
  • [16] Laura Gastaldo, Raphaèle Herbin, and Jean-Claude Latché, A discretization of the phase mass balance in fractional step algorithms for the drift-flux model, IMA J. Numer. Anal. 31 (2011), no. 1, 116-146. MR 2755939, https://doi.org/10.1093/imanum/drp006
  • [17] Vivette Girault and Pierre-Arnaud Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. MR 851383
  • [18] Dionysis Grapsas, Raphaèle Herbin, Walid Kheriji, and Jean-Claude Latché, An unconditionally stable staggered pressure correction scheme for the compressible Navier-Stokes equations, SMAI J. Comput. Math. 2 (2016), 51-97. MR 3633545
  • [19] Jean-Luc Guermond, Some implementations of projection methods for Navier-Stokes equations, RAIRO Modél. Math. Anal. Numér. 30 (1996), no. 5, 637-667. MR 1411394
  • [20] F. H. Harlow and A. A. Amsden, Numerical calculation of almost incompressible flow, J. Comput. Phys. 3 (1968), 80-93.
  • [21] F. H. Harlow and A. A. Amsden, A numerical fluid dynamics calculation method for all flow speeds, J. Comput. Phys. 8 (1971), 197-213.
  • [22] 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
  • [23] Walid Kheriji, Raphaèle Herbin, and Jean-Claude Latché, Pressure correction staggered schemes for barotropic one-phase and two-phase flows, Comput. & Fluids 88 (2013), 524-542. MR 3131211, https://doi.org/10.1016/j.compfluid.2013.09.022
  • [24] R. Herbin, W. Kheriji, and J.-C. Latché, On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 6, 1807-1857. MR 3342164, https://doi.org/10.1051/m2an/2014021
  • [25] R. Herbin and J.-C. Latché, Kinetic energy control in the MAC discretization of the compressible Navier-Stokes equations, Int. J. Finite Vol. 7 (2010), no. 2, 6. MR 2753586
  • [26] R. Herbin, J.-C. Latché, and T. T. Nguyen, Explicit staggered schemes for the compressible Euler equations, Applied mathematics in Savoie--AMIS 2012: Multiphase flow in industrial and environmental engineering, ESAIM Proc., vol. 40, EDP Sci., Les Ulis, 2013, pp. 83-102. MR 3095649, https://doi.org/10.1051/proc/201340006
  • [27] ISIS.
    A CFD computer code for the simulation of reactive turbulent flows.

    https://gforge.irsn.fr/gf/project/isis.
  • [28] B. Larrouturou, How to preserve the mass fractions positivity when computing compressible multi-component flows, J. Comput. Phys. 95 (1991), no. 1, 59-84. MR 1112315, https://doi.org/10.1016/0021-9991(91)90253-H
  • [29] Pierre-Louis Lions, Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996. MR 1422251
  • [30] Chun Liu and Noel J. Walkington, Convergence of numerical approximations of the incompressible Navier-Stokes equations with variable density and viscosity, SIAM J. Numer. Anal. 45 (2007), no. 3, 1287-1304 (electronic). MR 2318813, https://doi.org/10.1137/050629008
  • [31] Yohei Morinishi, Skew-symmetric form of convective terms and fully conservative finite difference schemes for variable density low-Mach number flows, J. Comput. Phys. 229 (2010), no. 2, 276-300. MR 2565600, https://doi.org/10.1016/j.jcp.2009.09.021
  • [32] F. Nicoud, Conservative high-order finite-difference schemes for low-Mach number flows, J. Comput. Phys. 158 (2000), no. 1, 71-97. MR 1743333, https://doi.org/10.1006/jcph.1999.6408
  • [33] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Methods Partial Differential Equations 8 (1992), no. 2, 97-111. MR 1148797, https://doi.org/10.1002/num.1690080202
  • [34] F. Schieweck and L. Tobiska, A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation, RAIRO Modél. Math. Anal. Numér. 23 (1989), no. 4, 627-647. MR 1025076
  • [35] F. Schieweck and L. Tobiska, An optimal order error estimate for an upwind discretization of the Navier-Stokes equations, Numer. Methods Partial Differential Equations 12 (1996), no. 4, 407-421. MR 1396464, https://doi.org/10.1002/(SICI)1098-2426(199607)12:4$ \langle $407::AID-NUM1$ \rangle $3.0.CO;2-Q
  • [36] 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, https://doi.org/10.1137/0729004
  • [37] 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, https://doi.org/10.1007/BF01396220

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 35Q30, 35Q55, 65N12, 76M10, 76M12

Retrieve articles in all journals with MSC (2010): 35Q30, 35Q55, 65N12, 76M10, 76M12


Additional Information

J. C. Latché
Affiliation: Institut de Radioprotection et de Sûreté Nucléaire (IRSN), France
Email: jean-claude.latche@irsn.fr

K. Saleh
Affiliation: Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 bd 11 novembre 1918, F-69622 Villeurbanne cedex, France
Email: saleh@math.univ-lyon1.fr

DOI: https://doi.org/10.1090/mcom/3241
Keywords: Variable density, Navier-Stokes equations, staggered schemes, analysis, convergence
Received by editor(s): February 24, 2014
Received by editor(s) in revised form: February 21, 2015, February 29, 2016, and October 28, 2016
Published electronically: August 7, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society