Encyclopedia of Applied and Computational Mathematics

2015 Edition
| Editors: Björn Engquist

Rosenbrock Methods

  • Florian Augustin
  • Peter Rentrop
Reference work entry
DOI: https://doi.org/10.1007/978-3-540-70529-1_143


Generalized Runge-Kutta methods; Linear-implicit Runge-Kutta methods; Rosenbrock methods; SDIRK methods


Rosenbrock methods are suitable for the numerical solution of stiff initial value problems
$$\displaystyle{ y^{\prime} = f(x,y),\quad y(x_{0}) = y_{0},\quad y \in \mathbb{R}^{n}. }$$
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  1. 1.
    Alexander, R.: Diagonally implicit Runge-Kutta methods for stiff O.D.E.’s. SIAM J. Numer. Anal. 14, 1006–1021 (1977)Google Scholar
  2. 2.
    Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Wiley, Chichester/New York (1987)zbMATHGoogle Scholar
  3. 3.
    Gottwald, B.A., Wanner, G.: A reliable Rosenbrock integrator for stiff differential systems. Comput. 26, 335–360 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Günther, M., Hoschek, M., Rentrop, P.: Differential-algebraic equations in electric circuit simulation. Int. J. Electron. Commun. 54, 101–107 (2000)Google Scholar
  5. 5.
    Günther, M., Kvaerno, A., Rentrop, P.: Multirate partitioned Runge-Kutta methods. BIT 41, 504–514 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hairer, E., Lubich, C., Roche, M.: The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Springer, Berlin/Heidelberg (1989)zbMATHGoogle Scholar
  7. 7.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer, Berlin/New York (1991)CrossRefzbMATHGoogle Scholar
  8. 8.
    van der Houwen, P.J.: Construction of Integration Formulas for Initial Value Problems. North Holland, Amsterdam (1977)zbMATHGoogle Scholar
  9. 9.
    Kaps, P., Rentrop, P.: Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differfential equations. Numer. Math. 33, 55–68 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kaps, P., Wanner, G.: A study of Rosenbrock-type methods of high order. Numer. Math. 38, 279–298 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nørsett, S.P., Wolfbrandt, A.: Order conditions for Rosenbrock type methods. Numer. Math. 32, 1–15 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes. Cambridge University Press, New York (1996)zbMATHGoogle Scholar
  13. 13.
    Rentrop, P.: Partitioned Runge-Kutta methods with stiffness detection and stepsize control. Numer. Math. 47, 545–564 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rosenbrock, H.H.: Some general implicit processes for the numerical solution of differential equations. Comput. J. 5, 329–330 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shampine, L.F., Reichelt, M.W.: The MATLAB ODE suite. SIAM J. Sci. Comput. 18, 1–22 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Steihaug, T., Wolfbrandt, A.: An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations. Math. Comput. 33, 521–534 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Steinebach, G., Rentrop, P.: An adaptive method of lines approach for modeling flow and transport in rivers. In: van de Wouwer et al. (eds.) Adaptive Method of Lines, pp. 181–205. Chapman Hall/CRC, Boca Raton (2001)Google Scholar
  18. 18.
    Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  19. 19.
    Strehmel, K., Weiner, R.: Linear-implizite Runge-Kutta Methoden und ihre Anwendung. Teubner Verlag, Stuttgart-Leibzig (1992)CrossRefzbMATHGoogle Scholar
  20. 20.
    Veldhuizen, M.: D-stability and Kaps-Rentrop methods. Computing 32, 229–237 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Florian Augustin
    • 1
  • Peter Rentrop
    • 1
  1. 1.Technische Universität MünchenFakultät MathematikMunichGermany