Advertisement

Journal of Scientific Computing

, Volume 29, Issue 2, pp 201–217 | Cite as

A Spectral Method for the Time Evolution in Parabolic Problems

  • A. Y. Suhov
Article

Abstract

Numerical time propagation of linear parabolic problems is commonly performed by Taylor expansion based schemes, such as Runge–Kutta. However, explicit schemes of this type impose a stringent stability restriction on the time step when the space discretization matrix is poorly conditioned. Thus the computational work required for integration over a long and fixed time interval is controlled by stability rather than by accuracy of the scheme. We develop an improved time evolution scheme based on a new Chebyshev series expansion for solving time-dependent inhomogeneous parabolic initial-boundary value problems in which the stability condition is relaxed. Spectral accuracy of the time evolution scheme is achieved. Additionally, the approximation derived here can be useful for solving quasi-linear parabolic evolution problems by exponential time differencing methods

Keywords

Spectral methods explicit scheme parabolic problems Chebyshev expansion 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz M., Stegun I.A. (1972). Handbook of Mathematical Functions. Dover, NYMATHGoogle Scholar
  2. 2.
    Amos D.E., Burgmeier J.W. (1973). Computation with three-term, linear, nonhomogeneous recursion relations. SIAM Rev 15:335–351CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Beylkin G., Keiser J.M., Vozovoi L. (1998). A new class of time discretization schemes for the solution of nonlinear PDEs. J. Comp. Phys 147:362–387CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Clenshaw C.W., Curtis A.R. (1960). A method for numerical integration on an automatic computer. Numer. Math 2: 197–205CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Cox S.M., Matthews P.C. (2002). Exponential time differencing for stiff systems. J. Comp. Phys 176:430–455CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Friedman A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, NJ.Google Scholar
  7. 7.
    Gardiner C.W. (1990). Handbook of Stochastic Methods. Springer-Verlag, NYMATHGoogle Scholar
  8. 8.
    Gottlieb D., Orszag S.A. (1977). Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, PhiladelphiaMATHGoogle Scholar
  9. 9.
    Gottlieb D., Turkel E. (1983). Spectral Methods for Time-Dependent Partial Differential Equations. In: Brezzi F (eds). Lecture Notes in Mathematics Vol 1127. Springer, Berlin, pp. 115–155Google Scholar
  10. 10.
    Gottlieb D., Lustman L. (1983). The spectrum of the Chebyshev collocation operator for the heat equation. SIAM J. Numer. Anal 20: 909–921CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Holcman, D., and Schuss, Z. Modelling calcium dynamics in dendritic spines. SIAM J. Appl. Math. (in press).Google Scholar
  12. 12.
    Meinardus G. (1967). Approximation of Functions: Theory and Numerical Methods. Springer-Verlag, NYMATHGoogle Scholar
  13. 13.
    Trefethen L.N., Trummer M.R. (1987). An instability phenomenon in spectral methods. SIAM J. Numer. Anal 24:1008–1023CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Trefethen L.N. (1999). Computation of pseudospectra. Acta Numerica 8: 247–295MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Rainville E.D. (1967). Special Functions. The Macmillan Company, NYMATHGoogle Scholar
  16. 16.
    Schuss Z. (1980). Theory and Applications of Stochastic Differential Equations. Wiley, NYMATHGoogle Scholar
  17. 17.
    Tal-Ezer H. (1989). Spectral methods in time for parabolic problems. SIAM J. Numer. Anal 26:1–11CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Thacher H.C. (1964). Conversion of a power to a series of Chebyshev polynomials. Comm. ACM 7: 181–182CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Toh K.-C., Trefethen L.N. (1999). The Kreiss matrix theorem on a general complex domain. SIAM J. Matrix Anal. Appl 21: 145–165CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Wilmott P., Dewynne J., Howison S. (1995). Option Pricing, Mathematical Models and Computation, students ed., Oxford Financial PressGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Applied MathematicsTel-Aviv UniversityTel-AvivIsrael

Personalised recommendations