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An unconditionally stable pressure correction schemefor the compressible barotropic Navier-Stokes equations

Published online by Cambridge University Press:  27 March 2008

Thierry Gallouët ,
Laura Gastaldo ,
Raphaele Herbin  and
Jean-Claude Latché
Thierry Gallouët
Affiliation:
Université de Provence, France. gallouet@cmi.univ-mrs.fr
Laura Gastaldo
Affiliation:
Institut de Radioprotection et de Sûreté Nucléaire (IRSN), France. laura.gastaldo@irsn.fr
Raphaele Herbin
Affiliation:
Université de Provence, France. herbin@cmi.univ-mrs.fr
Jean-Claude Latché
Affiliation:
Institut de Radioprotection et de Sûreté Nucléaire (IRSN), France. jean-claude.latche@irsn.fr
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Abstract

We present in this paper a pressure correction scheme for the barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based a priori estimates of the continuous problem also hold for the discrete solution.The stability proof is based on two independent results for general finite volume discretizations, both interesting for their own sake: the L 2-stability of the discrete advection operator provided it is consistent, in some sense, with the mass balance and the estimate of the pressure work by means of the time derivative of the elastic potential.The proposed scheme is built in order to match these theoretical results, and combines a fractional-step time discretization of pressure-correction type with a space discretization associating low order non-conforming mixed finite elements and finite volumes.Numerical tests with an exact smooth solution show the convergence of the scheme.

Keywords

Compressible Navier-Stokes equationspressure correction schemes.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 2 , March 2008 , pp. 303 - 331
Copyright
© EDP Sciences, SMAI, 2008

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References

Angot, P., Dolejší, V., Feistauer, M. and Felcman, J., Analysis of a combined barycentric finite volume-nonconforming finite element method for nonlinear convection-diffusion problems. Appl. Math. 4 (1998) 263–310. CrossRef
Bijl, H. and Wesseling, P., A unified method for computing incompressible and compressible flows in boundary-fitted coordinates. J. Comp. Phys. 141 (1998) 153–173. CrossRef
Bristeau, M.O., Glowinski, R., Dutto, L., Périaux, J. and Rogé, G., Compressible viscous flow calculations using compatible finite element approximations. Internat. J. Numer. Methods Fluids 11 (1990) 719–749. CrossRef
Casulli, V. and Greenspan, D., Pressure method for the numerical solution of transient, compressible fluid flows. Internat. J. Numer. Methods Fluids 4 (1984) 1001–1012. CrossRef
Chorin, A.J., Numerical solution of the Navier-Stokes equations. Math. Comp. 22 (1968) 745–762. CrossRef
P.G. Ciarlet, Finite elements methods – Basic error estimates for elliptic problems, in Handbook of Numerical Analysis II, P. Ciarlet and J.-L. Lions Eds., North Holland (1991) 17–351.
Colella, P. and Pao, K., A projection method for low speed flows. J. Comp. Phys. 149 (1999) 245–269. CrossRef
Crouzeix, M. and Raviart, P.-A., Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. RAIRO Anal. Numér. 7 (1973) 33–75.
K. Deimling, Nonlinear Functional Analysis. Springer, New-York (1980).
Demirdžić, I., Ž. Lilek and M. Perić, A collocated finite volume method for predicting flows at all speeds. Internat. J. Numer. Methods Fluids 16 (1993) 1029–1050. CrossRef
Dolejší, V., Feistauer, M., Felcman, J. and Kliková, A., Error estimates for barycentric finite volumes combined with nonconforming finite elements applied to nonlinear convection-diffusion problems. Appl. Math. 47 (2002) 301–340. CrossRef
A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences 159. Springer (2004).
Eymard, R., Gallouët, T., Ghilani, M. and Herbin, R., Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal. 18 (1998) 563–594. CrossRef
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis VII, P. Ciarlet and J.-L. Lions Eds., North Holland (2000) 713–1020.
E. Feireisl, Dynamics of viscous compressible flows, Oxford Lecture Series in Mathematics and its Applications 6. Oxford University Press (2004).
H. Feistauer, J. Felcman and I. Straškraba, Mathematical and computational methods for compressible flows, Oxford Science Publications. Clarendon Press (2003).
Fortin, M., Manouzi, H. and Soulaimani, A., On finite element approximation and stabilization methods for compressible viscous flows. Internat. J. Numer. Methods Fluids 17 (1993) 477–499. CrossRef
Guermond, J.-L. and Quartapelle, L., A projection FEM for variable density incompressible flows. J. Comp. Phys. 165 (2000) 167–188. CrossRef
Guermond, J.-L., Minev, P. and Shen, J., An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg. 195 (2006) 6011–6045. CrossRef
Harlow, F.H. and Amsden, A.A., Numerical calculation of almost incompressible flow. J. Comp. Phys. 3 (1968) 80–93. CrossRef
Harlow, F.H. and Amsden, A.A., A numerical fluid dynamics calculation method for all flow speeds. J. Comp. Phys. 8 (1971) 197–213. CrossRef
Issa, R.I., Solution of the implicitly discretised fluid flow equations by operator splitting. J. Comp. Phys. 62 (1985) 40–65. CrossRef
Issa, R.I. and Javareshkian, M.H., Pressure-based compressible calculation method utilizing total variation diminishing schemes. AIAA J. 36 (1998) 1652–1657. CrossRef
Issa, R.I., Gosman, A.D. and Watkins, A.P., The computation of compressible and incompressible recirculating flows by a non-iterative implicit scheme. J. Comp. Phys. 62 (1986) 66–82. CrossRef
Karki, K.C. and Patankar, S.V., Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations. AIAA J. 27 (1989) 1167–1174. CrossRef
Kobayashi, M.H. and Pereira, J.C.F., Characteristic-based pressure correction at all speeds. AIAA J. 34 (1996) 272–280. CrossRef
Larrouturou, B., How to preserve the mass fractions positivity when computing compressible multi-component flows. J. Comp. Phys. 95 (1991) 59–84. CrossRef
P.-L. Lions, Mathematical topics in fluid mechanics, Volume 2: Compressible models, Oxford Lecture Series in Mathematics and its Applications 10, Oxford University Press (1998).
M. Marion and R. Temam, Navier-Stokes equations: Theory and approximation, in Handbook of Numerical Analysis VI, P. Ciarlet and J.-L. Lions Eds., North Holland (1998).
Moukalled, F. and Darwish, M., A high-resolution pressure-based algorithm for fluid flow at all speeds. J. Comp. Phys. 168 (2001) 101–133. CrossRef
Nithiarasu, P., Codina, R. and Zienkiewicz, O.C., The Characteristic-Based Split (CBS) scheme – a unified approach to fluid dynamics. Internat. J. Numer. Methods Engrg. 66 (2006) 1514–1546. CrossRef
A. Novotný and I. Straškraba, Introduction to the mathematical theory of compressible flow, Oxford Lecture Series in Mathematics and its Applications 27. Oxford University Press (2004).
Patnaik, G., Guirguis, R.H., Boris, J.P. and Oran, E.S., A barely implicit correction for flux-corrected transport. J. Comp. Phys. 71 (1987) 1–20. CrossRef
Politis, E.S. and Giannakoglou, K.C., A pressure-based algorithm for high-speed turbomachinery flows. Internat. J. Numer. Methods Fluids 25 (1997) 63–80. 3.0.CO;2-A>CrossRef
Rannacher, R. and Turek, S., Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differential Equations 8 (1992) 97–111. CrossRef
Temam, R., Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires II. Arch. Rat. Mech. Anal. 33 (1969) 377–385. CrossRef
van der Heul, D.R., Vuik, C. and Wesseling, P., Stability analysis of segregated solution methods for compressible flow. Appl. Numer. Math. 38 (2001) 257–274. CrossRef
van der Heul, D.R., Vuik, C. and Wesseling, P., A conservative pressure-correction method for flow at all speeds. Comput. Fluids 32 (2003) 1113–1132. CrossRef
Van Dormaal, J.P., Raithby, G.D. and McDonald, B.H., The segregated approach to predicting viscous compressible fluid flows. Trans. ASME 109 (1987) 268–277.
Vidović, D., Segal, A. and Wesseling, P., A superlinearly convergent Mach-uniform finite volume method for the Euler equations on staggered unstructured grids. J. Comput. Phys. 217 (2006) 277–294. CrossRef
Wall, C., Pierce, C.D. and Moin, P., A semi-implicit method for resolution of acoustic waves in low Mach number flows. J. Comp. Phys. 181 (2002) 545–563. CrossRef
Wenneker, I., Segal, A. and Wesseling, P., Mach-uniform, A unstructured staggered grid method. Internat. J. Numer. Methods Fluids 40 (2002) 1209–1235. CrossRef
P. Wesseling, Principles of computational fluid dynamics, Springer Series in Computational Mathematics 29. Springer (2001).
Zienkiewicz, O.C. and Codina, R., A general algorithm for compressible and incompressible flow – Part I. The split characteristic-based scheme. Internat. J. Numer. Methods Fluids 20 (1995) 869–885. CrossRef