Advertisement

Numerische Mathematik

, Volume 102, Issue 2, pp 277–291 | Cite as

Fast Runge-Kutta approximation of inhomogeneous parabolic equations

  • María López-Fernández
  • Christian LubichEmail author
  • Cesar Palencia
  • Achim Schädle
Article

Abstract

The result after N steps of an implicit Runge-Kutta time discretization of an inhomogeneous linear parabolic differential equation is computed, up to accuracy ɛ, by solving only

Open image in new window

linear systems of equations. We derive, analyse, and numerically illustrate this fast algorithm.

Mathematics Subject Classification (2000)

65M20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ashyralyev, A., Sobolevskii, P: Well-Posedness of Parabolic Difference Equations. Birkhäuser, Basel, 1994Google Scholar
  2. 2.
    Bakaev, N.Y., Thomée, V., Wahlbin, L.: Maximum-norm estimates for resolvents of elliptic finite element operators. Math. Comp. 72, 1597–1610 (2002)MathSciNetGoogle Scholar
  3. 3.
    Brenner, P., Crouzeix, M., Thomée, V.: Single step methods for inhomogeneous linear differential equations in Banach space. RAIRO Modél. Math. Anal. Numér. 16, 5–26 (1982)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Cavers, I.A.: A hybrid tridiagonalization algorithm for symmetric sparse matrices. SIAM J. Matrix Anal. Appl. 15, 1363–1380 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Gavrilyuk, I.P., Makarov, V.: Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces. Preprint, 2004Google Scholar
  6. 6.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems. Second edition. Springer, Berlin, 1996Google Scholar
  7. 7.
    Lewis, R.W., Morgan, K., Thomas, H.R., Seetharamu, K.N.: The Finite Element Method in Heat Transfer Analysis. John Wiley & Sons Ltd, Chichester, 1996Google Scholar
  8. 8.
    López-Fernández, M., Palencia, C.: On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51, 289-303 (2004)zbMATHMathSciNetGoogle Scholar
  9. 9.
    López-Fernández, M., Palencia, C., Schädle, A.: A spectral order method for inverting sectorial Laplace transforms. Preprint, 2005Google Scholar
  10. 10.
    Lubich, C., Ostermann, A.: Runge-Kutta methods for parabolic equations and convolution quadrature. Math. Comp. 60, 105–131 (1993)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Lubich, C., Schädle, A.: Fast convolution for nonreflecting boundary conditions. SIAM J. Sci. Comp. 24, 161–182 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Nikolajsen, J.L.: An improved Laguerre eigensolver for unsymmetric matrices. SIAM J. Sci. Comp. 22, 822–834 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Schädle, A., López-Fernández, M., Lubich, C.: Fast and oblivious convolution quadrature. Preprint, 2005Google Scholar
  14. 14.
    Sheen, D., Sloan, I.H., Thomée, V.: A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature. Math. Comp. 69, 177–195 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Sheen, D., Sloan, I.H., Thomée, V.: A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature. IMA J. Numer. Anal. 23, 269–299 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Shewchuk, J.R.: Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator. In: Applied Computational Geometry: Towards Geometric Engineering, M.C. Lin, D. Manocha (eds.), Lecture Notes in Computer Science 1148, Springer, 1996, pp. 203–222Google Scholar
  17. 17.
    Stenger, F.: Approximations via Whittaker's cardinal function. J. Approx. Theory 17, 222–240 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Stenger, F.: Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Review 23, 165–224 (1981)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Talbot, A.: The accurate numerical inversion of Laplace transforms. J. Inst. Math. Appl. 23, 97–120 (1979)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • María López-Fernández
    • 1
  • Christian Lubich
    • 2
    Email author
  • Cesar Palencia
    • 1
  • Achim Schädle
    • 3
  1. 1.Departamento de Matemática Aplicada y ComputaciónUniversidad de ValladolidValladolidSpain
  2. 2.Mathematisches InstitutUniversität TübingenTübingenGermany
  3. 3.ZIB BerlinBerlinGermany

Personalised recommendations