Skip to main content
SpringerLink
Log in

Applications of a variational method for mixed differential equations

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

Summary

Variational principles for elliptic boundary-value problems as well as linear initial-value problems have been derived by various investigators. For initial-value problems Tonti and Reddy have used a convolution type of bilinear form of the functional for the time-like coordinate. This introduces a certain amount of directionality thereby reflecting the initial-value nature of the problem. In the present investigation the methods of Tonti and Reddy are used to derive the appropriate variational formulation for the transonic flow problem. A number of linear and non-linear examples have been investigated. As a test for the existence of directionality, finite-differences are used to discretize the variational integral. For initial-value problems of wave equation and diffusion equation type, fully implicit finite-difference approximations are recovered. The small-disturbance transonic equation leads to the Murman and Cole differencing theory; when applied to the full potential-flow equations, the rotated difference scheme due to Jameson is obtained.

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. Friedrichs K.O., Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958) 333–418.

    Google Scholar 

  2. Katsanis T., Numerical solution of symmetric positive differential equations, Math. Comp. 22 (1968) 763–783.

    Google Scholar 

  3. Lesaint P., Finite element methods for symmetric hyperbolic equations, Numer. Math. 21 (1973) 244–255.

    Google Scholar 

  4. Aziz, A. K. and Leventhal, S. H., Numerical solutions of equations of hyperbolic-elliptic type, (1974) Naval Ordnance Laboratory NOLTR 74–144.

  5. Filippov A., On difference methods for the solution of the Tricomi problem, Izv. Acad. Nauk SSR. Ser. Mat. 2 (1957) 73–88.

    Google Scholar 

  6. Ogawa H., On difference methods for the solution of a Tricomi problem, Trans. Amer. Math. Soc. 100 (1961) 404–424.

    Google Scholar 

  7. Murman E. M. and Cole J. D., Calculation of plane steady transonic flows, AIAA J. 9 (1971) 114–121.

    Google Scholar 

  8. Jameson, A., Numerical calculation of three-dimensional transonic flow over a yawed wing, Proceedings of AIAA Computational Fluid Dynamics Conference, Palm Springs, Calif. (1973) 18–26.

  9. Gurtin M. E., Variational principles for linear initial-value problems, Quart. Appl. Math. 22 (1964) 252–256.

    Google Scholar 

  10. Tonti E., Annal. Mat. Pura Appl. 95 (1973) 331–359.

    Google Scholar 

  11. Reddy J. N., A note of mixed variational principles for initial-value problems, Quart. J. Mech. Appl. Math. 28 (1975) 123–132.

    Google Scholar 

  12. Vainberg M. M., Variational methods for the study of non-linear operators, translated by A. Feinstein; Holden-Day, San Fransisco (1964).

    Google Scholar 

  13. Atherton R. W. and Homsy G. M., On the existence and formulation of variational principles for non-linear differential equations, Studies in Appl. Mathematics 54 (1975) 31–60.

    Google Scholar 

  14. Khosla, P. K. and Rubin, S. G., Applications of a variational method for mixed differential equations, (1980), University of Cincinnati Report No. AFL 80-10-55.

  15. Finlayson B. A., The method of weighted residuals and variational principles, Academic Press, New York (1972).

    Google Scholar 

  16. Rasmussen H., Applications of variational methods in compressible flow calculations, Progress in Aerospace Sciences 15 (1974) 1–35; Edited by Küchemann, Pergamon Press.

    Google Scholar 

  17. Geffen N., Yaniv S, A note on the variational formulation of quasi-linear initial-value problems, ZAMP 27 (1976) 833–838.

    Google Scholar 

  18. Bateman H., Irrotational motion of a compressible inviscid fluid, Proc. Nat. Acad. Sci. 16 (1930) 816–825.

    Google Scholar 

  19. Lush P. E. and Cherry T. M., The variational method in hydrodynamics. Quart. J. Mech. Appl. Math. 9 (1956) 6–21.

    Google Scholar 

  20. Greenspan D. and Jain P., Application of a method for approximating extremals of functionals to compressible subsonic fluid flows, J. Math. Anal. Appl. 19 (1967) 85–111.

    Google Scholar 

  21. Miller K. S., An introduction to the calculus of finite differences and difference equations, Henry Holt and Company, New York (1960).

    Google Scholar 

  22. von Mises R., Mathematical theory of compressible fluid flow, Academic Press, New York (1958).

    Google Scholar 

  23. Rubin, S. G. and Khosla, P. K., An integral spline method for boundary-layer equations, Polytechnic Institute of New York Report No. POLY M/AE 77-12, July (1977).

Download references

Author information

Authors and Affiliations

  1. Department of Aerospace Engineering and Applied Mechanics, University of Cincinnati, 45221, Cincinnati, Ohio, 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

An extended version of this paper was first presented at the Bat-Sheva International Seminar on Finite Elements for Non-Elliptic Problems, Tel-Aviv, Israel, July 1977.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khosla, P.K., Rubin, S.G. Applications of a variational method for mixed differential equations. J Eng Math 15, 185–200 (1981). https://doi.org/10.1007/BF00042779

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00042779

Keywords

Search

Navigation