Skip to main content
SpringerLink
Log in

Filtering of non-linear instabilities

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Summary

For Courant numbers larger than one and cell Reynolds numbers larger than two, oscillations and in some cases instabilities are typically found with implicit numerical solutions of the fluid dynamics equations. This behavior has sometimes been associated with the loss of diagonal dominance of the coefficient matrix. It is shown here that these problems can in fact be related to the choice of the spatial differences, with the resulting instability related to aliasing or non-linear interaction. Appropriate “filtering” can reduce the intensity of these oscillations and in some cases possibly eliminate the instability. These filtering procedures are equivalent to a weighted average of conservation and non-conservation differencing. The entire spectrum of filtered equations retains a three-point character as well as second-order spatial accuracy. Burgers equation has been considered as a model. Several filters are examined in detail, and smooth solutions have been obtained for extremely large cell Reynolds numbers.

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. Patrick J. Roache, Computational fluid dynamics. Hermosa Publishers, 1972.

  2. P. K. Khosla and S. G. Rubin, A diagonally dominant second order accurate implicit scheme. Computers and Fluids 2 (1974) 207–209.

    Google Scholar 

  3. David H. Rudy and Richard S. Hirsch, Comments on the role of diagonal dominance in implicit methods. NASA TM X-73905, May 1976.

  4. N. A. Phillips; An example of non-linear computational instability. The Atmosphere and Sea in Motion, pp. 501–504, Rockefeller Inst. Press, New York (1959).

    Google Scholar 

  5. A. Arakawa, Computational design for long term numerical integration of the equations of fluid motion, two dimensional incompressible flow, Part. I. J. Comput. Physics 1 (1966) 199–243.

    Google Scholar 

  6. Steve A. Piacsek and G. P. Williams, Conservation properties of convention difference schemes. J. of Comput. Physics 6 (1970) 392–405.

    Google Scholar 

  7. F. G. Shuman, Numerical methods in weather prediction: II smoothing and filtering. Monthly Weather Review 85, 11 (1957) 357–361.

    Google Scholar 

  8. R. Shapiro, Smoothing, filtering and boundary effects. Reviews in Geophysics and Space Physics 8 (1970) 359–387.

    Google Scholar 

  9. Sin-I Cheng and G. Shubin, Computational accuracy and mesh Reynolds number. Department of Aerospace and Mechanical Sciences, Princeton University Report NOOO14-75-C-0376-003 July 1976.

  10. Frank P. Taverna Jr. and Richard J. Busch Jr., Burgers equation with smoothing. Class Project, Polytechnic Institute of New York.

Download references

Author information

Authors and Affiliations

  1. Aerodynamics Laboratories, Polytechnic Institute of New York, 11735, Farmingdale, New York, USA

    P. K. Khosla & S. G. Rubin

Authors
  1. P. K. Khosla
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. G. Rubin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was sponsored by the National Aeronautics and Space Administration, Langley Research Center, Hampton, Va., under Grant NSG-1244

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khosla, P.K., Rubin, S.G. Filtering of non-linear instabilities. J Eng Math 13, 127–141 (1979). https://doi.org/10.1007/BF00042748

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00042748

Keywords

Search

Navigation