Skip to main content
SpringerLink
Log in

Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We present a review of the semi-Lagrangian method for advection-diffusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable.

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. Loft, R. D., Thomas, S. J., and Dennis, J. M. (2001). Terascale spectral element dynamical core for atmospheric general circulation models. InProceedings of Supercomuting 2001, Denver.

  2. Maday, Y., Patera, A. T., and Ronquist, E. M. (1990). An operator-integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow.J. Sci. Comp. 4, 263–292.

    Article  MathSciNet  Google Scholar 

  3. Xiu, D., and Karniadakis, G. E. (2001). A semi-Lagrangian high-order method for Navier-Stokes equations.J. Comp. Phys. 172, 658.

    Article  MATH  MathSciNet  Google Scholar 

  4. Robert, A. (1981). A stable numerical integration scheme for the primitive meteorological equations.Atmos. Ocean 19, 35.

    Google Scholar 

  5. Pironneau, O. (1982). On the transport-diffusion algorithm and its applications to the Navier-Stokes equations.Numer. Math. 38, 309.

    Article  MATH  MathSciNet  Google Scholar 

  6. Giraldo, F. X. (2003). Strong and weak Lagrange-Galerkin spectral element methods for shallow water equations.Comput. Math. Appl. 45, 97–121.

    Article  MATH  MathSciNet  Google Scholar 

  7. Karniadakis, G. E., and Orszag, S. A. (1993). Nodes, modes, and flow codes.Phys. Today 34.

  8. Karniadakis, G. E., and Sherwin, S. J. (1999).Spectral/hpElement Methods for CFD, Oxford University Press, London.

    MATH  Google Scholar 

  9. Falcone, M., and Ferretti, R. (1998). Convergence analysis for a class of high-order semi-Lagrangian advection schemes.SIAM J. Numer. Anal. 35, 909.

    Article  MATH  MathSciNet  Google Scholar 

  10. Huffenus, J. P., and Khaletzky, D. (1984). A finite element method to solve the Navier-Stokes equations using the method of characteristics.Int. J. Numer. Methods Fluids 4, 247.

    Article  MATH  Google Scholar 

  11. Malevsky, A. V., and Thomas, S. J. (1997). Parallel algorithms for semi-Lagrangian advection.Int. J. Numer. Methods Fluids 25, 455.

    Article  MATH  Google Scholar 

  12. Oliveira, A., and Baptista, A. M. (1995). A comparison of integration and interpolation Eulerian-Lagrangian methods.Int. J. Numer. Methods Fluids 21, 183.

    Article  MATH  MathSciNet  Google Scholar 

  13. Staniforth, A., and Cote, J. (1991). Semi-Lagrangian integration schemes for atmospheric models — a review.Mon. Wea. Rev. 119, 2206.

    Article  Google Scholar 

  14. Malevsky, A. V. (1996). Spline-characteristic method for simulation of convective turbulence.J. Comput. Phys. 123, 466.

    Article  MATH  Google Scholar 

  15. Bartello, P., and Thomas, S. J. (1996). The cost-effectiveness of semi-Lagrangian advection.Mon. Wea. Rev. 124, 2883.

    Article  Google Scholar 

  16. Giraldo, F. X. (1998). The Lagrange-Galerkin spectral element method on unstructured quadrilateral grids.J. Comput. Phys. 147, 114.

    Article  MATH  MathSciNet  Google Scholar 

  17. McGregor, J. L. (1993). Economical determination of departure points for semi-Lagrangian models.Mon. Wea. Rev. 121, 221.

    Article  Google Scholar 

  18. McDonald, A., and Bates, J. R. (1987). Improving the estimate of the departure point position in a two-time level semi-Lagrangian and semi-implicit scheme.Mon. Wea. Rev. 115, 737.

    Article  Google Scholar 

  19. McDonald, A. (1984). Accuracy of multi-upstream, semi-Lagrangian advective schemes.Mon. Wea. rev. 112, 1267.

    Article  Google Scholar 

  20. Suli, E. S. (1988). Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations.Numer. Math. 53, 459.

    Article  MathSciNet  Google Scholar 

  21. Allievi, A., and Bermejo, R. (2000). Finite element modified method of characteristics for the Navier-Stokes equations.Int. J. Numer. Methods Fluids 32, 439.

    Article  MATH  Google Scholar 

  22. Priestley, A. (1994). Exact projections and the Lagrange-Galerkin method: A realistic alternative to quadrature.J. Comput. Phys. 112, 316.

    Article  MATH  MathSciNet  Google Scholar 

  23. Achdou, Y., and Guermond, J. L. (2000). Convergence analysis of a finite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations.SIAM J. Numer. Anal. 37, 799.

    Article  MATH  MathSciNet  Google Scholar 

  24. Deville, M. O., Fischer, P. F., and Mund, E. H. (2002).High-Order Methods for Incompressible Fluid Flow. Cambridge University Press.

  25. Momeni-Masuleh, S. H. (2001).Spectral Methods for the Three Field Formulation of Incompressible Fluid Flow. PhD thesis, Aberystwyth.

  26. Sherwin, S. J. (2003). A substepping Navier-Stokes splitting scheme for spectral/hp element discretisations. In Matuson, T., Ecer, A., Periaux, J., Satufka, N., and Fox, P. (eds.),Parallel Computational Fluid Dynamics: New Frontiers Multi-Disciplinary Applications, North-Holland, pp. 43–52.

  27. Leriche, E., and Labrosse, G. (2000). High-order direct Stokes solvers with or without temporal splitting: numerical investigations of their comparative properties.SIAM J. Sci. Comput. 22(4), 1386.

    Article  MATH  MathSciNet  Google Scholar 

  28. Ghia, U., Ghia, K. N., and Shin, C. T. (1982). High-Re solutions for the incompressible flow using the Navier-Stokes equations and a multigrid method.J. Comput. Phys. 48, 387.

    Article  MATH  Google Scholar 

  29. Koseff, J. R., and Street, R. L. (1984). The lid-driven cavity flow: a synthesis of qualitative and quantitative observations.ASME J. Fluids Eng. 106, 390.

    Article  Google Scholar 

  30. Xu, J., Xiu, D., and Karniadakis, G. E. (2002). A semi-Lagrangian method for turbulence simulations using mixed spectral discretizations.J. Sci. Comput. 17, 585.

    Article  MATH  MathSciNet  Google Scholar 

  31. Smolarkiewicz, P. K., and Pudykiewicz, J. (1992). A class of semi-Lagrangian approximations for fluids.J. Atmo. Sci. 49, 2082.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, USA

    D. Xiu, S. Dong & G. E. Karniadakis

  2. Department of Aeronautics, Imperial College, USA

    S. J. Sherwin

Authors
  1. D. Xiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. J. Sherwin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G. E. Karniadakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xiu, D., Sherwin, S.J., Dong, S. et al. Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows. J Sci Comput 25, 323–346 (2005). https://doi.org/10.1007/BF02728994

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02728994

Key words

Search

Navigation