Journal of Scientific Computing

, Volume 15, Issue 1, pp 79–116 | Cite as

On Error Bounds of Finite Difference Approximations to Partial Differential Equations—Temporal Behavior and Rate of Convergence

  • Saul Abarbanel
  • Adi Ditkowski
  • Bertil Gustafsson
Article

Abstract

This paper considers a family of spatially semi-discrete approximations, including boundary treatments, to hyperbolic and parabolic equations. We derive the dependence of the error-bounds on time as well as on mesh size.

finite difference error bounds 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    Abarbanel, S., and Ditkowski, A. (1997). Multi-dimensional asymptotically stable 4th-order accurate schemes for the diffusion equation. ICASE Report No. 96-8, February 1996. Also, Asymptotically stable fourth-order accurate schemes for the diffusion equation on complex shapes. J. Comput. Phys. 133(2), 279–288 (1997).Google Scholar
  2. 2.
    Abarbanel, S., and Ditkowski, A. (1999). Multi-dimensional asymptotically stable schemes for advection-diffusion equations. ICASE Report 47-96. Also Computers and Fluids 28, 481–510.Google Scholar
  3. 3.
    Ditkowski, A. (1997). Bounded-Error Finite Difference Schemes for Initial Boundary Value Problems on Complex Domains. Thesis, Department of Applied Mathematics, School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel.Google Scholar
  4. 4.
    Ditkowski, A. Fourth-Order Bounded-Error Schemes for the Variable Coefficients Diffusion Equation in Complex Geometries, in preparation.Google Scholar
  5. 5.
    Ditkowski, A., Dridi, K. H., and Hesthaven, J. S. (1999). Convergent cartesian grid methods for Maxwells equations in complex geometries. J. Comput. Phys., submitted.Google Scholar
  6. 6.
    Gustafsson, G., Kreiss, H. O., and Sundström, A. (1972). Stability theory of difference approximations for mixed initial boundary value problems. II. Math. Comp. 26, 649–686.Google Scholar
  7. 7.
    Gustafsson, B., Kreiss, H. O., and Oliger, J. (1995). Time Dependent Problems and Difference Methods, Wiley.Google Scholar
  8. 8.
    Goganov, S. K., and Ryabenkii, V. S. (1963). Spectral criteria for the stability of bound-ary-value problems for non-self-adjoint difference equations. Uspeki Mat. 18 VIII, 3–15.Google Scholar
  9. 9.
    Gustafsson, B. (1975). The convergence rate for difference approximations to mixed initial boundary value problems. Math. Comp. 29, 296–406.Google Scholar
  10. 10.
    Gustafsson, B. (1981). The convergence rate for difference approximations to general mixed initial boundary value problems. SIAM J. Numer. Amal. 18(2), 179–190.Google Scholar
  11. 11.
    Kreiss, H. O. (1966). Difference approximations for the initial boundary value problem for hyperbolic differential equations. In Greenspan, D. (ed.), Numerical Solutions of Nonlinear Partial Differential Equations, Wiley, New York.Google Scholar
  12. 12.
    Kreiss, H. O. (1968). Stability theory for difference approximations of mixed initial boundary value problem. I. Math. Comp. 22, 703–714.Google Scholar
  13. 13.
    Kreiss, H. O., and Scherer, G. (1974). Finite element and finite difference methods for hyperbolic partial differential equations. In Mathematical Aspects of Finite Element in Partial Differential Equations, Academic Press, Inc.Google Scholar
  14. 14.
    Kreiss, H. O., and Scherer, G. (1977). On the Existence of Energy Estimates for Difference Approximations for Hyperbolic Systems. Technical report, Department of Scientific Computing, Uppsala University.Google Scholar
  15. 15.
    Olsson, P. (1995). Summation by parts, projections and stability. I. Math. of Comp. 64(211), 1035–1065.Google Scholar
  16. 16.
    Olsson, P. (1995). Summation by parts, projections and stability. II. Math. of Comp. 64(212), 1473–1493.Google Scholar
  17. 17.
    Osher, S. (1969). Systems of difference equations with general homogeneous boundary conditions. Tran. Amer. Math. Soc. 137, 177–201.Google Scholar
  18. 18.
    Strand, B. (1994). Summation by parts for finite difference approximations for d-dx. J. Comput. Phys. 110, 47–67.Google Scholar
  19. 19.
    Strand, B. (1996). High Order Difference Approximations for Hyperbolic Initial boundary Value Problems. Thesis, Department of Scientific Computing, Uppsala University, Uppsala, Sweden.Google Scholar
  20. 20.
    Strikwerda, J. C. (1980). Initial boundary value problems for method of lines. J. Comput. Phys. 34, 94–110.Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Saul Abarbanel
    • 1
  • Adi Ditkowski
    • 2
  • Bertil Gustafsson
    • 3
  1. 1.School of Mathematical Sciences, Department of Applied MathematicsTel-Aviv UniversityTel-AvivIsrael
  2. 2.Division of Applied MathematicsBrown UniversityProvidence
  3. 3.Department of Scientific ComputingUppsala UniversityUppsalaSweden

Personalised recommendations