Low Mach number limit of some staggered schemes for compressible barotropic flows

Herbin, R.; Latché, J.-C.; Saleh, K.

doi:10.1090/mcom/3604

Low Mach number limit of some staggered schemes for compressible barotropic flows
HTML articles powered by AMS MathViewer

by R. Herbin, J.-C. Latché and K. Saleh;

Math. Comp. 90 (2021), 1039-1087

DOI: https://doi.org/10.1090/mcom/3604

Published electronically: February 4, 2021

HTML | PDF | Request permission

Abstract:

In this paper, we study the behaviour of some staggered discretization based numerical schemes for the barotropic Navier-Stokes equations at low Mach number. Three time discretizations are considered: the implicit-in-time scheme and two non-iterative pressure correction schemes. The two latter schemes differ by the discretization of the convection term: linearly implicit for the first one, so that the resulting scheme is unconditionally stable, and explicit for the second one, so that the scheme is stable under a CFL condition involving the material velocity only. We prove rigorously that these three variants are asymptotic preserving in the following sense: for a given mesh and a given time step, a sequence of solutions obtained with a sequence of vanishing Mach numbers tends to a solution of a standard scheme for incompressible flows. This convergence result is obtained by mimicking the proof of convergence of the solutions of the (continuous) barotropic Navier-Stokes equations to that of the incompressible Navier-Stokes equation as the Mach number vanishes. Numerical results performed with a hand-built analytical solution show the behaviour that is expected from the analysis. Additional numerical results are obtained for the shock solutions of problems which are not in the scope of the present non dimensionalization but are nevertheless interesting to understand the behaviour of the scheme.

References

Thomas Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal. 180 (2006), no. 1, 1–73. MR 2211706, DOI 10.1007/s00205-005-0393-2
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
A. Arakawa and V. Lamb, A potential enstrophy and energy conserving scheme for the shallow water equations, Monthly Weather Review 109 (1981), 18–36.
Hester Bijl and Pieter Wesseling, A unified method for computing incompressible and compressible flows in boundary-fitted coordinates, J. Comput. Phys. 141 (1998), no. 2, 153–173. MR 1619651, DOI 10.1006/jcph.1998.5914
CALIF$^3$S, A software components library for the computation of reactive turbulent flows, https://gforge.irsn.fr/gf/project/isis.
V. Casulli and D. Greenspan, Pressure method for the numerical solution of transient, compressible fluid flows, Internat. J. Numer. Methods Fluids 4 (1984), 1001–1012.
Christophe Chalons, Mathieu Girardin, and Samuel Kokh, An all-regime Lagrange-projection like scheme for the gas dynamics equations on unstructured meshes, Commun. Comput. Phys. 20 (2016), no. 1, 188–233. MR 3514275, DOI 10.4208/cicp.260614.061115a
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
Phillip Colella and Karen Pao, A projection method for low speed flows, J. Comput. Phys. 149 (1999), no. 2, 245–269. MR 1672739, DOI 10.1006/jcph.1998.6152
Floraine Cordier, Pierre Degond, and Anela Kumbaro, An asymptotic-preserving all-speed scheme for the Euler and Navier-Stokes equations, J. Comput. Phys. 231 (2012), no. 17, 5685–5704. MR 2947989, DOI 10.1016/j.jcp.2012.04.025
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
Raphaël Danchin, Zero Mach number limit in critical spaces for compressible Navier-Stokes equations, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 1, 27–75 (English, with English and French summaries). MR 1886005, DOI 10.1016/S0012-9593(01)01085-0
Pierre Degond and Min Tang, All speed scheme for the low Mach number limit of the isentropic Euler equations, Commun. Comput. Phys. 10 (2011), no. 1, 1–31. MR 2775032, DOI 10.4208/cicp.210709.210610a
Stéphane Dellacherie, Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number, J. Comput. Phys. 229 (2010), no. 4, 978–1016. MR 2576236, DOI 10.1016/j.jcp.2009.09.044
Benoit Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1986, 2271–2279. MR 1702718, DOI 10.1098/rspa.1999.0403
B. Desjardins, E. Grenier, P.-L. Lions, and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl. (9) 78 (1999), no. 5, 461–471. MR 1697038, DOI 10.1016/S0021-7824(99)00032-X
Giacomo Dimarco, Raphaël Loubère, and Marie-Hélène Vignal, Study of a new asymptotic preserving scheme for the Euler system in the low Mach number limit, SIAM J. Sci. Comput. 39 (2017), no. 5, A2099–A2128. MR 3704264, DOI 10.1137/16M1069274
David G. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force, Ann. of Math. (2) 105 (1977), no. 1, 141–200. MR 431261, DOI 10.2307/1971029
David G. Ebin, Motion of slightly compressible fluids in a bounded domain. I, Comm. Pure Appl. Math. 35 (1982), no. 4, 451–485. MR 657824, DOI 10.1002/cpa.3160350402
R. Eymard, T. Gallouët, R. Herbin, and J.-C. Latché, Convergence of the MAC scheme for the compressible Stokes equations, SIAM J. Numer. Anal. 48 (2010), no. 6, 2218–2246. MR 2763662, DOI 10.1137/090779863
E. Feireisl, Asymptotic analysis of the full Navier-Stokes-Fourier system: from compressible to incompressible fluid flows, Russian Math. Surveys 62 (2007), 511–533.
Eduard Feireisl and Antonín Novotný, The low Mach number limit for the full Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal. 186 (2007), no. 1, 77–107. MR 2338352, DOI 10.1007/s00205-007-0066-4
Eduard Feireisl, Antonín Novotný, and Hana Petzeltová, Low Mach number limit for the Navier-Stokes system on unbounded domains under strong stratification, Comm. Partial Differential Equations 35 (2010), no. 1, 68–88. MR 2748618, DOI 10.1080/03605300903279377
Isabelle Gallagher, Résultats récents sur la limite incompressible, Astérisque 299 (2005), Exp. No. 926, vii, 29–57 (French, with French summary). Séminaire Bourbaki. Vol. 2003/2004. MR 2167201
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
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
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
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
H. Guillard, Recent developments in the computation of compressible low Mach flows, Flow, Turbulence and Combustion 76 (2006), 363–369.
Hervé Guillard, On the behavior of upwind schemes in the low Mach number limit. IV: P0 approximation on triangular and tetrahedral cells, Comput. & Fluids 38 (2009), no. 10, 1969–1972. MR 2645814, DOI 10.1016/j.compfluid.2009.06.003
Hervé Guillard and Cécile Viozat, On the behaviour of upwind schemes in the low Mach number limit, Comput. & Fluids 28 (1999), no. 1, 63–86. MR 1651839, DOI 10.1016/S0045-7930(98)00017-6
Jeffrey Haack, Shi Jin, and Jian-Guo Liu, An all-speed asymptotic-preserving method for the isentropic Euler and Navier-Stokes equations, Commun. Comput. Phys. 12 (2012), no. 4, 955–980. MR 2913446, DOI 10.4208/cicp.250910.131011a
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
Raphaèle Herbin, Jean-Claude Latché, and Trung Tan Nguyen, Consistent segregated staggered schemes with explicit steps for the isentropic and full Euler equations, ESAIM Math. Model. Numer. Anal. 52 (2018), no. 3, 893–944. MR 3865553, DOI 10.1051/m2an/2017055
R. I. Issa, Solution of the implicitly discretised fluid flow equations by operator-splitting, J. Comput. Phys. 62 (1986), no. 1, 40–65. MR 825890, DOI 10.1016/0021-9991(86)90099-9
R. I. Issa, A. D. Gosman, and A. P. Watkins, The computation of compressible and incompressible recirculating flows by a noniterative implicit scheme, J. Comput. Phys. 62 (1986), no. 1, 66–82. MR 825891, DOI 10.1016/0021-9991(86)90100-2
K.C. Karki and S.V. Patankar, Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations, AIAA Journal 27 (1989), 1167–1174.
Sergiu Klainerman and Andrew Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34 (1981), no. 4, 481–524. MR 615627, DOI 10.1002/cpa.3160340405
Heinz-Otto Kreiss, Problems with different time scales for partial differential equations, Comm. Pure Appl. Math. 33 (1980), no. 3, 399–439. MR 562742, DOI 10.1002/cpa.3160330310
Nipun Kwatra, Jonathan Su, Jón T. Grétarsson, and Ronald Fedkiw, A method for avoiding the acoustic time step restriction in compressible flow, J. Comput. Phys. 228 (2009), no. 11, 4146–4161. MR 2524514, DOI 10.1016/j.jcp.2009.02.027
J.-C. Latché, B. Piar, and K. Saleh, A discrete kinetic energy preserving convection operator for variable density flows on locally refined staggered meshes, Submitted, hal-02391939 (2018).
J. C. Latché and K. Saleh, A convergent staggered scheme for the variable density incompressible Navier-Stokes equations, Math. Comp. 87 (2018), no. 310, 581–632. MR 3739211, DOI 10.1090/mcom/3241
P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9) 77 (1998), no. 6, 585–627 (English, with English and French summaries). MR 1628173, DOI 10.1016/S0021-7824(98)80139-6
A. Majda and J. Sethian, The derivation and numerical solution of the equations for zero Mach number combustion, Combustion Science and Technology 42 (1985), 185–205.
J.J. McGuirk and G.J. Page, Shock capturing using a pressure-correction method, AIAA Journal 28 (1990), 1751–1757.
G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal. 158 (2001), no. 1, 61–90. MR 1834114, DOI 10.1007/PL00004241
Yann Moguen, Tarik Kousksou, Pascal Bruel, Jan Vierendeels, and Erik Dick, Pressure-velocity coupling allowing acoustic calculation in low Mach number flow, J. Comput. Phys. 231 (2012), no. 16, 5522–5541. MR 2942695, DOI 10.1016/j.jcp.2012.05.001
S. Noelle, G. Bispen, K. R. Arun, M. Lukáčová-Medviďová, and C.-D. Munz, A weakly asymptotic preserving low Mach number scheme for the Euler equations of gas dynamics, SIAM J. Sci. Comput. 36 (2014), no. 6, B989–B1024. MR 3293458, DOI 10.1137/120895627
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
Steven Schochet, Hyperbolic-hyperbolic singular limits, Comm. Partial Differential Equations 12 (1987), no. 6, 589–632. MR 879353, DOI 10.1080/03605308708820504
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
E. Turkel, Preconditioning techniques in computational fluid dynamics, Annual review of fluid mechanics, Vol. 31, Annu. Rev. Fluid Mech., vol. 31, Annual Reviews, Palo Alto, CA, 1999, pp. 385–416. MR 1670946, DOI 10.1146/annurev.fluid.31.1.385
D. R. van der Heul, C. Vuik, and P. Wesseling, Stability analysis of segregated solution methods for compressible flow, Appl. Numer. Math. 38 (2001), no. 3, 257–274. MR 1847066, DOI 10.1016/S0168-9274(01)00028-9
D. R. van der Heul, C. Vuik, and P. Wesseling, A conservative pressure-correction method for flow at all speeds, Comput. & Fluids 32 (2003), no. 8, 1113–1132. MR 1966263, DOI 10.1016/S0045-7930(02)00086-5
J.P. Van Dormaal, G.D. Raithby, and B.H. McDonald, The segregated approach to predicting viscous compressible fluid flows, Transactions of the ASME 109 (1987), 268–277.
D. Vidović, A. Segal, and P. Wesseling, A superlinearly convergent Mach-uniform finite volume method for the Euler equations on staggered unstructured grids, J. Comput. Phys. 217 (2006), no. 2, 277–294. MR 2260602, DOI 10.1016/j.jcp.2006.01.031
Clifton Wall, Charles D. Pierce, and Parviz Moin, A semi-implicit method for resolution of acoustic waves in low Mach number flows, J. Comput. Phys. 181 (2002), no. 2, 545–563. MR 1927401, DOI 10.1006/jcph.2002.7141
I. Wenneker, A. Segal, and P. Wesseling, A Mach-uniform unstructured staggered grid method, Internat. J. Numer. Methods Fluids 40 (2002), no. 9, 1209–1235. MR 1939062, DOI 10.1002/fld.417
S.Y. Yoon and T. Yabe, The unified simulation for incompressible and compressible flow by the predictor-corrector scheme based on the CIP method, Comput. Phys. Commun. 119 (1999), 149–158.
Hamed Zakerzadeh, On the Mach-uniformity of the Lagrange-projection scheme, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 4, 1343–1366. MR 3702416, DOI 10.1051/m2an/2016064