Convergence of the MAC scheme for the compressible stationary Navier-Stokes equations

Gallouët, T.; Herbin, R.; Latché, J.-C.; Maltese, D.

doi:10.1090/mcom/3260

Convergence of the MAC scheme for the compressible stationary Navier-Stokes equations
HTML articles powered by AMS MathViewer

by T. Gallouët, R. Herbin, J.-C. Latché and D. Maltese PDF

Math. Comp. 87 (2018), 1127-1163 Request permission

Abstract:

We prove in this paper the convergence of the marker and cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two- or three-dimensional Cartesian grids. Existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the approximate solutions, up to a subsequence, and in an appropriate sense. We then prove that the limit of the approximate solutions satisfies the mass and momentum balance equations, as well as the equation of state, which is the main difficulty of this study.

References

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
V. Casulli and D. Greenspan, Pressure method for the numerical solution of transient, compressible fluid flows, Internat. J. Numer. Methods Fluids 4 (1984), no. 11, 1001–1012.
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
Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
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 10.1086/phos.67.4.188705
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
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
R. Eymard, T. Gallouët, R. Herbin, and J. C. Latché, A convergent finite element-finite volume scheme for the compressible Stokes problem. II. The isentropic case, Math. Comp. 79 (2010), no. 270, 649–675. MR 2600538, DOI 10.1090/S0025-5718-09-02310-2
Eduard Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 26, Oxford University Press, Oxford, 2004. MR 2040667
Eduard Feireisl and Antonín Novotný, Singular limits in thermodynamics of viscous fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. MR 2499296, DOI 10.1007/978-3-7643-8843-0
A. Fettah and T. Gallouët, Numerical approximation of the general compressible Stokes problem, IMA J. Numer. Anal. 33 (2013), no. 3, 922–951. MR 3081489, DOI 10.1093/imanum/drs024
A. Fettah, T. Gallouët, and H. Lakehal, An existence proof for the stationary compressible Stokes problem, Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), no. 4, 847–875 (English, with English and French summaries). MR 3270426, DOI 10.5802/afst.1427
T. Gallouët, R. Herbin, and J.-C. Latché, A convergent finite element-finite volume scheme for the compressible Stokes problem. I. The isothermal case, Math. Comp. 78 (2009), no. 267, 1333–1352. MR 2501053, DOI 10.1090/S0025-5718-09-02216-9
T. Gallouët, R. Herbin, and J.-C. Latché, $W^{1,q}$ stability of the Fortin operator for the MAC scheme, Calcolo 49 (2012), no. 1, 63–71. MR 2891275, DOI 10.1007/s10092-011-0045-x
T. Gallouët, R. Herbin, J.-C. Latché, and K. Mallem, Convergence of the MAC scheme for the incompressible Navier-Stokes equations on non-uniform grids, Found. Comput. Math., 2016.
Thierry Gallouët, Raphaèle Herbin, David Maltese, and Antonin Novotny, Error estimates for a numerical approximation to the compressible barotropic Navier-Stokes equations, IMA J. Numer. Anal. 36 (2016), no. 2, 543–592. MR 3483096, DOI 10.1093/imanum/drv028
F. Harlow and A. Amsden, Numerical calculation of almost incompressible flow, J. Comput. Phys. 3 (1968), no. 1, 80–93.
F. Harlow and A. Amsden, A numerical fluid dynamics calculation method for all flow speeds, J. Comput. Phys. 8 (1971), no. 2, 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
R. Herbin, J.-C. Latché, and N. Therme, Consistency of some staggered schemes for the Euler equations, in preparation.
David Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal. 132 (1995), no. 1, 1–14. MR 1360077, DOI 10.1007/BF00390346
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. Karki and S. Patankar, Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations, AIAA J. 27 (1989), no. 9, 1167–1174.
Kenneth H. Karlsen and Trygve K. Karper, A convergent mixed method for the Stokes approximation of viscous compressible flow, IMA J. Numer. Anal. 32 (2012), no. 3, 725–764. MR 2954728, DOI 10.1093/imanum/drq048
Trygve K. Karper, A convergent FEM-DG method for the compressible Navier-Stokes equations, Numer. Math. 125 (2013), no. 3, 441–510. MR 3117509, DOI 10.1007/s00211-013-0543-7
Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 2, Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press, Oxford University Press, New York, 1998. Compressible models; Oxford Science Publications. MR 1637634
Sébastien Novo and Antonin Novotný, On the existence of weak solutions to the steady compressible Navier-Stokes equations when the density is not square integrable, J. Math. Kyoto Univ. 42 (2002), no. 3, 531–550. MR 1967222, DOI 10.1215/kjm/1250283849
A. Novotný and I. Straškraba, Introduction to the mathematical theory of compressible flow, Oxford Lecture Series in Mathematics and its Applications, vol. 27, Oxford University Press, Oxford, 2004. MR 2084891
Denis Serre, Variations de grande amplitude pour la densité d’un fluide visqueux compressible, Phys. D 48 (1991), no. 1, 113–128 (French, with English summary). MR 1098658, DOI 10.1016/0167-2789(91)90055-E
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
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
Pieter Wesseling, Principles of computational fluid dynamics, Springer Series in Computational Mathematics, vol. 29, Springer-Verlag, Berlin, 2001. MR 1796357, DOI 10.1007/978-3-642-05146-3