Skip to main content
SpringerLink
Log in

Conditionally Stable Multidimensional Schemes for Advective Equations

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

Abstract

It is shown that the isotropic wave-like multidimensional spatial stencils combined with linear multistep and Runge-Kutta time marching schemes provide more favorable stability restrictions for advective initial-value problems. Under certain conditions the maximum allowable time step can be doubled compared to using conventional spatial stencils. Consequently, this paper shows that the multidimensional optimizations of spatial schemes, involving more grid points, are not inherently less efficient in terms of the processing time. Three numerical tests solving the two and three dimensional advection equations are carried out to experiment the stability restrictions found in the previous sections.

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. Abarbanel, S., Ditkowski, A.: Multi-dimensional asymptotically stable finite difference schemes for the advection-diffusion equation. Comput. Fluids 28, 481–510 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abarbanel, S., Gottlieb, D.: Stability of two-dimensional initial boundary value problems using leap-frog type schemes. Math. Comput. 33, 1145–1155 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baldauf, M.: Stability analysis of linear discretisations of the advection equation with Runge-Kutta time integration. J. Comput. Phys. 227, 6638–6659 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bogey, C., Bailly, C.: A family of low dispersive and low dissipative explicit schemes for flow and noise computation. J. Comput. Phys. 194, 194–214 (2004)

    Article  MATH  Google Scholar 

  5. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, New York (1986)

    Google Scholar 

  6. Godunov, S.K., Ryabenkii, V.S.: Special stability criteria of boundary value problems for non-selfadjoint difference equations. Russ. Math. Surv. 18, 1–12 (1963)

    Article  MATH  Google Scholar 

  7. Gulin, A.V., Yukhno, L.F.: Numerical investigation of the stability of two-layer difference schemes for the two-dimensional heat conduction equation. Comput Math. Math. Phys. 36, 1079–1085 (1996)

    MATH  MathSciNet  Google Scholar 

  8. Gustafsson, B., Kreiss, H.-O., Sundstrom, A.: Stability theory for difference approximations of mixed initial boundary value problems. II. Math. Comput. 26, 649–686 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hirsch, C.: Numerical Computation of Internal and External Flows. Fundamentals of Numerical Discretization, vol. 1. Wiley, New York (2001)

    Google Scholar 

  10. Hixon, R., Allampalli, V., Nallasamy, M., Sawyer, S.D.: High-accuracy large-step explicit Runge-Kutta (HALE-RK) schemes for computational aeroacoustics. AIAA Paper 2006-797 (2006)

  11. Hu, F.Q., Hussaini, M.Y., Manthey, J.L.: Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics. J. Comput. Phys. 124, 177–191 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kennedy, C.A., Carpenter, M.H.: Several new numerical methods for compressible shear-layer simulations. Appl. Num. Math. 14, 397–433 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kreiss, H.-O.: Stability theory for difference approximations of mixed initial boundary value problems. I. Math. Comput. 22, 703–714 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lax, P.D., Richtmyer, R.D.: Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. 9, 125–151 (1956)

    MathSciNet  Google Scholar 

  15. Levy, D., Tadmor, E.: From semi-discrete to fully-discrete stability of Runge-Kutta schemes by the energy method. SIAM Rev. 40, 1–27 (1998)

    Article  MathSciNet  Google Scholar 

  16. Michelson, D.: Stability theory of difference approximations for multidimensional initial-boundary-value problems. Math. Comput. 40, 1–45 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  17. Sescu, A., Hixon, R., Afjeh, A.A.: Multidimensional optimization of finite difference schemes for computational aeroacoustics. J. Comput. Phys. 227, 4563–4588 (2008)

    Article  MATH  Google Scholar 

  18. Smolarkiewicz, P.K.: The multi-dimensional Crowley advection scheme. Mon. Weather. Rev. 110, 1968-1983 (1982)

    Article  Google Scholar 

  19. Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations. SIAM, Philadelphia (2004)

    MATH  Google Scholar 

  20. Tam, C.K.W., Webb, J.C.: Dispersion-relation-preserving finite difference schemes for computational aeroacoustics. J. Comput. Phys. 107, 262–281 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  21. Trefethen, L.N.: Group velocity in finite difference schemes. SIAM Rev. 24, 113 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  22. Vichnevetsky, R., Bowles, J.B.: Fourier analysis of numerical approximations of hyperbolic equations, SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1982)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MIME Department, University of Toledo, 2801 W Bancroft, MS 312, Toledo, 43606, OH, USA

    Adrian Sescu, Abdollah A. Afjeh, Ray Hixon & Carmen Sescu

Authors
  1. Adrian Sescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdollah A. Afjeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ray Hixon
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Carmen Sescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adrian Sescu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sescu, A., Afjeh, A.A., Hixon, R. et al. Conditionally Stable Multidimensional Schemes for Advective Equations. J Sci Comput 42, 96 (2010). https://doi.org/10.1007/s10915-009-9317-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-009-9317-x

Keywords

Search

Navigation