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

Latché, J.; Saleh, K.

doi:10.1090/mcom/3241

A convergent staggered scheme for the variable density incompressible Navier-Stokes equations
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by J. C. Latché and K. Saleh;

Math. Comp. 87 (2018), 581-632

DOI: https://doi.org/10.1090/mcom/3241

Published electronically: August 7, 2017

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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

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, DOI 10.1002/fld.2270
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, DOI 10.1007/978-3-319-05684-5_{7}
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, DOI 10.1002/fld.3837
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
Susanne C. Brenner, Korn’s inequalities for piecewise $H^1$ vector fields, Math. Comp. 73 (2004), no. 247, 1067–1087. MR 2047078, DOI 10.1090/S0025-5718-03-01579-5
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 343661
Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
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, DOI 10.1016/j.jcp.2008.03.027
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, DOI 10.1007/BF01393835
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, DOI 10.1093/imanum/18.4.563
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, DOI 10.1016/S1570-8659(00)07005-8
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, DOI 10.1093/imanum/drn084
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, DOI 10.1051/m2an:2008005
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, DOI 10.3934/cpaa.2012.11.2371
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, DOI 10.1051/m2an/2010002
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, DOI 10.1093/imanum/drp006
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
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, DOI 10.5802/smai-jcm.9
Jean-Luc Guermond, Some implementations of projection methods for Navier-Stokes equations, RAIRO Modél. Math. Anal. Numér. 30 (1996), no. 5, 637–667 (English, with English and French summaries). MR 1411394, DOI 10.1051/m2an/1996300506371
F. H. Harlow and A. A. Amsden, Numerical calculation of almost incompressible flow, J. Comput. Phys. 3 (1968), 80–93.
F. H. Harlow and A. A. Amsden, A numerical fluid dynamics calculation method for all flow speeds, J. Comput. Phys. 8 (1971), 197–213.
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
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, DOI 10.1016/j.compfluid.2013.09.022
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, DOI 10.1051/m2an/2014021
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
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, DOI 10.1051/proc/201340006
ISIS. A CFD computer code for the simulation of reactive turbulent flows. https://gforge.irsn.fr/gf/project/isis.
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, DOI 10.1016/0021-9991(91)90253-H
Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 1, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996. Incompressible models; Oxford Science Publications. MR 1422251
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. MR 2318813, DOI 10.1137/050629008
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, DOI 10.1016/j.jcp.2009.09.021
F. Nicoud, Conservative high-order finite-difference schemes for low-Mach number flows, J. Comput. Phys. 158 (2000), no. 1, 71–97. MR 1743333, DOI 10.1006/jcph.1999.6408
R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Methods Partial Differential Equations 8 (1992), no. 2, 97–111. MR 1148797, DOI 10.1002/num.1690080202
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 (English, with French summary). MR 1025076, DOI 10.1051/m2an/1989230406271
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, DOI 10.1002/(SICI)1098-2426(199607)12:4<407::AID-NUM1>3.0.CO;2-Q
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 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