Skip to main content
SpringerLink
Log in

Stability of Preconditioned Finite Volume Schemes at Low Mach Numbers

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

A finite volume method for inviscid unsteady flows at low Mach numbers is studied. The method uses a preconditioning of the dissipation term within the numerical flux function only. It can be observed by numerical experiments that the preconditioned scheme combined with an explicit time integrator is unstable if the time step Δt does not satisfy the requirement to be \(\mathcal{O}(M^2)\) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to \(\Delta t = \mathcal{O}(M)\), M → 0, though producing unphysical results.

A comprehensive mathematical substantiation of this numerical phenomenon by means of a von Neumann stability analysis is presented, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M–2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. The theoretical results are afterwards confirmed by numerical experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Choi and C. L. Merkle, The application of preconditioning in viscous flows, J. Comput. Phys., 105 (1993), pp. 207–223.

    Google Scholar 

  2. I. Demirdžić, Ž. Lilek, and M. Perić, A collocated finite volume method for predicting flows at all speeds, Int. J. Numer. Methods Fluids, 16 (1993), pp. 1029–1050.

  3. O. Friedrich, Weighted essential non-oscillatory schemes for the interpolation of mean values on unstructured grids, J. Comput. Phys., 144 (1998), pp. 194–212.

    Google Scholar 

  4. H. Guillard and C. Viozat, On the behaviour of upwind schemes in the low Mach number limit, Comp. Fluids, 28 (1999), pp. 63–86.

  5. F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8(12) (1965), pp. 2182–2189.

    Google Scholar 

  6. L. Hoffmann and A. Meister, A compressible low Mach number scheme based on image processing and asymptotic analysis, Comput. Fluid Dynam. J., 9 (2001), pp. 231–242.

  7. C. Hu and C.-W. Shu, Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150 (1999), pp. 97–127.

    Google Scholar 

  8. J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Vol. 114 of Applied Mathematical Sciences, Springer, New York, Berlin, Heidelberg, 1996.

  9. S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic system with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), pp. 481–524.

    Google Scholar 

  10. R. Klein, Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: One-dimensional flow, J. Comput. Phys., 121 (1995), pp. 213–237.

    Google Scholar 

  11. R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister, L. Hoffmann, and T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics, J. Eng. Math., 9(1/4)(2) (2001), pp. 261–343.

    Google Scholar 

  12. R. Massjung, Numerical Schemes and Well-Posedness in Nonlinear Aeroelasticity, PhD thesis, RWTH Aachen, 2002.

  13. A. Meister, Asymptotic single and multiple scale expansions in the low Mach number limit, SIAM J. Appl. Math., 60(1) (1999), pp. 256–271.

  14. A. Meister, Analyse und Anwendung Asymptotik-basierter numerischer Verfahren zur Simulation reibungsbehafteter Strömungen in allen Mach-Zahlbereichen, Habilitation, Department of Mathematics, Universität Hamburg, 2001.

  15. A. Meister, Theoretical Investigation of the Lax-Friedrichs Scheme in the low Mach Number Limit, in Discrete Modelling and Discrete Algorithms in Continuum Mechanics, Logos, Berlin, 2001, pp. 177–186.

  16. A. Meister, Asymptotic Expansions and Numerical Methods in Computational Fluid Dynamics, in Industrial Mathematics and Statistics, J. C. Misra et al., eds., New Delhi, 2002.

  17. A. Meister, Asymptotic based preconditioning technique for low Mach number flows, Z. Angew. Math. Mech., 83(1) (2003), pp. 3–25.

  18. A. Meister, Viscous flow fields at all speeds: Analysis and numerical simulation, J. Appl. Math. Phys. (ZAMP), 54 (2003), pp. 1010–1049.

  19. S. V. Patankar and D. B. Spalding, A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat Mass Transfer, 15 (1972), pp. 1787–1805.

    Google Scholar 

  20. P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43 (1981), pp. 357–372.

  21. T. Schneider, N. Botta, K. J. Geratz, and R. Klein, Semi-implicit extension of a Godunov-type compressible solver based on low Mach number asymptotics II: Multi-dimensional, variable density zero Mach number flow, J. Comput. Phys., 155 (1999), pp. 248–286.

    Google Scholar 

  22. C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), pp. 439–471.

    Google Scholar 

  23. J.-S. Shuen, K.-H. Chen, and Y. Choi, A coupled implicit method for chemical non-equilibrium flows at all speeds, J. Comput. Phys., 106 (1993), pp. 306–318.

    Google Scholar 

  24. E. Turkel, Preconditioned methods for solving the incompressible and low speed compressible equations, J. Comput. Phys., 72 (1987), pp. 277–298.

    Google Scholar 

  25. E. Turkel and V. Vatsa, Choice of Variables and Preconditioning for Time Dependent Problems, AIAA Paper, 2003-3692 CP, 2003, AIAA 16th Computational Fluid Dynamics Conference, Orlando.

  26. B. van Leer, W.-T. Lee, and P. Roe, Characteristic Time-Stepping or Local Preconditioning of the Euler Equations, AIAA Paper, AIAA-91-1552, 1991.

  27. Y. Wada and M.-S. Liou, A Flux Splitting Scheme with High-Resolution and Robustness for Discontinuities, AIAA Paper, 94–0083, 1994.

  28. J. M. Weiss and W. A. Smith, Preconditioning applied to variable and constant density flows, AIAA J., 33(11) (1995), pp. 2050–2057.

  29. I. Wenneker, A. Segal, and P. Wesseling, A Mach-uniform unstructured staggered grid method, I, J. Numer. Methods Fluids, 40 (2002), pp. 1209–1235.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Kassel, Heinrich-Plett-Str. 40, D-34132, Kassel, Germany

    P. Birken & A. Meister

Authors
  1. P. Birken
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Meister
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to P. Birken or A. Meister.

Additional information

AMS subject classification (2000)

35L65, 35C20, 76G25

Rights and permissions

Reprints and permissions

About this article

Cite this article

Birken, P., Meister, A. Stability of Preconditioned Finite Volume Schemes at Low Mach Numbers. Bit Numer Math 45, 463–480 (2005). https://doi.org/10.1007/s10543-005-0009-0

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-005-0009-0

Key words

Search

Navigation