Abstract
Motivated by the work of Karper [29], we propose a numerical scheme to compressible Navier-Stokes system in spatial multi-dimension based on finite differences. The backward Euler method is applied for the time discretization, while a staggered grid, with continuity and momentum equations on different grids, is used in space. The existence of a solution to the implicit nonlinear scheme, strictly positivity of the numerical density, stability and consistency of the method for the whole range of physically relevant adiabatic exponents are proved. The theoretical part is complemented by computational results that are performed in two spatial dimensions.
Funding: The research of the authors leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement 320078. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.
References
[1] P. Angot, C.-H. Bruneau, and P. Fabrie, A penalization method to take into account obstacles in incompressible viscous flows, Numer. Math. 81 (1999), No. 4, 497-520.10.1007/s002110050401Search in Google Scholar
[2] G. Ansanay-Alex, F. Babik, J. C. Latché, and D. Vola, An L2-stable approximation of the Navier-Stokes convection operator for low-order non-conforming finite elements, Int. J. Numer. Meth. Fluids 66 (2011), No. 5, 555-580.10.1002/fld.2270Search in Google Scholar
[3] Y. Coudiére, T. Gallouét, and R. Herbin, Discrete Sobolev inequalities and Lp error estimates for finite volume solutions of convection diffusion equations, ESAIM: M2AN35 (2001), 767-778.10.1051/m2an:2001135Search in Google Scholar
[4] P. I. Crumpton, J. A. Mackenzie, and K. W. Morton, Cell vertex algorithms for the compressible Navier-Stokes equations, J. Comput. Phys. 109 (1993), No. 1,1-15.10.1006/jcph.1993.1194Search in Google Scholar
[5] P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equations, Commun. Comput. Phys. 10 (2011), 1-31.10.4208/cicp.210709.210610aSearch in Google Scholar
[6] R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathemati-cae 98 (1989), No. 3, 511-547.10.1007/BF01393835Search in Google Scholar
[7] V. Dolejsí and M. Feistauer, Discontinuous Galerkin Method: Analysis and Applications to Compressible Flow, Springer Series in Computational Mathematics, Vol. 48, Springer, Cham, 2015.10.1007/978-3-319-19267-3Search in Google Scholar
[8] P. Drábekand R. Hosek, Properties of solution diagrams for bistable equations, Electron. J. Differ. Equ. 2015 (2015), No. 156, 1-19.Search in Google Scholar
[9] L. C. Evans, Partial Differential Equations, Providence, RI: American Mathematical Society, 1998.Search in Google Scholar
[10] R. Eymard, T. Gallouét, and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis. Vol. 7: Solution of Equations in R” (Part 3). Techniques of Scientific Computing (Part 3), Amsterdam: North-Holland, Elsevier, 2000, pp. 7131020.10.1016/S1570-8659(00)07005-8Search in Google Scholar
[11] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford, Oxford University Press, 2004.10.1093/acprof:oso/9780198528388.001.0001Search in Google Scholar
[12] E. Feireisl, P. Gwiazda, A. Swierczewska-Gwiazda, and E. Wiedemann, Dissipative measure-valued solutions to the compressible Navier-Stokes system, Calculus of Variations and Partial Differential Equations 55 (2016), No. 6, Paper No. 141, 20.10.1007/s00526-016-1089-1Search in Google Scholar
[13] E. Feireisl, R. Hosek, D. Maltese, and A. Novotny, Error estimates for a numerical method for the compressible Navier-Stokes system on sufficiently smooth domains, ESAIM: M2AN 51 (2017), No. 1, 279-319.10.1051/m2an/2016022Search in Google Scholar
[14] E. Feireisl, R. Hosek, and M. Michálek, A convergent numerical method for the full Navier-Stokes-Fourier system in smooth physical domains, SIAMJ. Numer. Anal. 54 (2016), No. 5, 3062-3082.10.1137/15M1011809Search in Google Scholar
[15] E. Feireisl, T. Karper, and M. Michálek, Convergence of a numerical method for the compressible Navier-Stokes system on general domains, Numer. Math. 134 (2016), No. 4, 667-704.10.1007/s00211-015-0786-6Search in Google Scholar
[16] E. Feireisl, T. Karper, and A. Novotny, A convergent numerical method for the Navier-Stokes-Fourier system, IMA J. Numer.Anal. 36 (2016), No. 4, 1477-1535.10.1093/imanum/drv049Search in Google Scholar
[17] E. Feireisl and M. Lukácová-Medvid’ová, Convergence of a mixed finite element finite volume scheme for the isentropic Navier-Stokes system via dissipative measure-valued solutions, Found. Comput. Math. 18 (2018), No. 3, 707-730.10.1007/s10208-017-9351-2Search in Google Scholar
[18] E. Feireisl, M. Lukácová-Medvid’ová, S. Necasová, A. Novotny, and B. She, Error Estimate for a Numerical Approximation to the Compressible Barotropic Navier-Stokes Equations, Preprint, 2016.Search in Google Scholar
[19] E. Feireisl, A. Novotny, and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech. 3 (2001), No. 4, 358-392.10.1007/PL00000976Search in Google Scholar
[20] T. Gallouét, L. Gastaldo, R. Herbin, and J.-C. Latché, An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations, ESAIM: M2AN42 (2008), 303-331.10.1051/m2an:2008005Search in Google Scholar
[21] T. Gallouét, R. Herbin, J.-C. Latché, and D. Maltese, Convergence of the MAC scheme for the compressible stationary Navier-Stokes equations, ArXive-prints (2016).10.1090/mcom/3260Search in Google Scholar
[22] T. Gallouét, R. Herbin, D. Maltese, and A. Novotny, Implicit MAC scheme for compressible Navier-Stokes equations: unconditional error estimates, Preprint (2016).10.1007/s00211-018-1007-xSearch in Google Scholar
[23] T. Gallouét, R. Herbin, D. Maltese, and A. Novotny, Convergence of the marker-and-cell scheme for the semi-stationary compressible Stokes problem, Mathematics and Computers in Simulation (2016), available on line.10.1016/j.matcom.2016.10.003Search in Google Scholar
[24] T. Gallouét, R. Herbin, D. Maltese, and A. Novotny, Error estimates for a numerical approximation to the compressible barotropic Navier-Stokes equations, IMA J. Numer. Anal. 36 (2016), No. 2, 543-592.10.1093/imanum/drv028Search in Google Scholar
[25] R. Glowinski, T.-W. Pan, and J. Periaux, A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations, Comput. Methods Appl. Mech. Eng. 112 (1994), No. 1-4,133-148.10.1016/0045-7825(94)90022-1Search in Google Scholar
[26] F. Grasso and C. Meola, Euler and Navier-Stokes Equations for Compressible Flows: Finite-Volume Methods, Handbook of Computational Fluid Mechanics, Academic Press, San Diego, CA, 1996, pp. 159-282.10.1016/B978-012553010-1/50005-0Search in Google Scholar
[27] J. Haack, S. Jin, and J.-G. Liu, An all-speed asymptotic-preserving method for the isentropic Euler and Navier-Stokes equations, Commun. Comput. Phys. 12 (2012), 955-980.10.4208/cicp.250910.131011aSearch in Google Scholar
[28] R. Hosek, Expressing the remainder of Taylor polynomial when the function lacks smoothness, Elem. Math. 72 (2017), No. 3, 126-130.10.4171/EM/335Search in Google Scholar
[29] T. K. Karper, A convergent FEM-DG method for the compressible Navier-Stokes equations, Numer. Math. 125 (2013), No. 3, 441-510.10.1007/s00211-013-0543-7Search in Google Scholar
[30] T. K. Karper, Convergent finite differences for 1D viscous isentropic flow in Eulerian coordinates, Discrete Contin. Dyn.Syst., Ser. S 7 (2014), No. 5, 993-1023.10.3934/dcdss.2014.7.993Search in Google Scholar
[31] K. Khadra, P. Angot, S. Parneix, and J.-P. Caltagirone, Fictiuous domain approach for numerical modelling of Navier-Stokes equations, Int. J. Numer. Methods Fluids 34 (2000), No. 8, 651-684.10.1002/1097-0363(20001230)34:8<651::AID-FLD61>3.0.CO;2-DSearch in Google Scholar
[32] C. M. Klaij, J. J. W. van der Vegt, and H. van der Ven, Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations, J. Comput. Phys. 217 (2006), No. 2, 589-611.10.1016/j.jcp.2006.01.018Search in Google Scholar
[33] M. Kouhi and E. Oñate, An implicit stabilized finite element method for the compressible Navier-Stokes equations using finite calculus, Comput. Mech. 56 (2015), No. 1,113-129.10.1007/s00466-015-1161-2Search in Google Scholar
[34] M. Kupiainen and B. Sjogreen, A Cartesian embedded boundary method for the compressible Navier-Stokes equations, J.Sci. Comput. 41 (2009), No.1, 94-117.10.1007/s10915-009-9289-xSearch in Google Scholar
[35] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models, Oxford, Clarendon Press, 1998.Search in Google Scholar
[36] B. Liu, The analysis of a finite element method with streamline diffusion for the compressible Navier-Stokes equations, SIAMJ. Numer. Anal. 38 (2000), No. 1,1-16 (electronic).10.1137/S0036142998336424Search in Google Scholar
[37] R. Mittal and G. laccarino, Immersed Boundary Methods, Annual Review of Fluid Mechanics, Vol. 37, Palo Alto, CA: Annual Reviews, 2005, pp. 239-261.10.1146/annurev.fluid.37.061903.175743Search in Google Scholar
[38] S. Noelle, G. Bispen, K. R. Arun, M. Lukácová-Medvid’ová, 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-B1024.10.1137/120895627Search in Google Scholar
[39] J. S. Park and C. Kim, Higher-order multi-dimensional limiting strategy for discontinuous Galerkin methods in compressible inviscid and viscous flows, Comput. & Fluids 96 (2014), 377-396.10.1016/j.compfluid.2013.11.030Search in Google Scholar
[40] C. S. Peskin, The immersed boundary method, Acta Numerica 11 (2002), 479-517.10.1017/S0962492902000077Search in Google Scholar
[41] F. Renac, S. Gérald, C. Marmignon, and F. Coquel, Fast time implicit-explicit discontinuous Galerkin method for the compressible Navier-Stokes equations, J. Comput. Phys. 251 (2013), 272-291.10.1016/j.jcp.2013.05.043Search in Google Scholar
[42] J. F. Thompson, Z. U. A. Warsi, and C. W. Mastin, Boundary-fitted coordinate systems for numerical solution of partial differential equations - a review, J. Comput. Phys. 47 (1982), No.1,1-108.10.1016/0021-9991(82)90066-3Search in Google Scholar
[43] A. Valli, An existence theorem for compressible viscous fluids, Ann. Mat. Pura Appl. (4) 130 (1982), 197-213.10.1007/BF01761495Search in Google Scholar
[44] K. Xu, C. Kim, L. Martinelli, and A. Jameson, BGK-based schemes for the simulation of compressible flow, Int. J. Comput.Fluid Dyn. 7 (1996), No. 3, 213-235.10.1080/10618569608940763Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston