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Perturbed collocation and Runge-Kutta methods

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Summary

It is well known thatsome implicit Runge-Kutta methods are equivalent to collocation methods. This fact permits very short and natural proofs of order andA, B, AN, BN-stability properties for this subclass of methods (see [9] and [10]). The present paper answers the natural question, ifall RK methods can be considered as a somewhat “perturbed” collocation. After having introduced this notion we give a proof on the order of the method and discuss their stability properties. Much of known theory becomes simple and beautiful.

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References

  1. Burrage, K.: High order algebraically stable Runge-Kutta methods BIT18, 373–383 (1978)

    Google Scholar 

  2. Burrage, K.: A special family of Runge-Kutta methods for solving stiff differential equations. BIT18, 22–41 (1978)

    Google Scholar 

  3. Butcher, J.C.: Integration processes based on Radau quadrature formulas. Math. Comput.18, 233–244 (1964)

    Google Scholar 

  4. Butcher, J.C., Burrage, K.: Stability criteria for implicit Runge-Kutta methods. SIAM J. Numer. Anal.16, 46–57 (1979)

    Google Scholar 

  5. Chipman, F.H.: A note on implicitA-stableR-K methods with parameters. BIT16, 223–227 (1976)

    Google Scholar 

  6. Dieudonné, J.: Foundations of modern analysis. AP 1960, 1969

  7. Ehle, B.L.: On Padé approximations to the exponential function andA-stable methods for the numerical solution of initial value problems. Research Report CSRR2010, 1969, Waterloo, Canada, see also BIT8, 276–278 (1968) and SIAM J. Math. Anal.4, 671–680 (1973)

  8. Nørsett, S.P.:C-polynomials for rational approximations to the exponential function. Numer. Math.25, 39–56 (1975)

    Google Scholar 

  9. Nørsett, S.P., Wanner, G.: The real-pole sandwich for rational approximations and oscillation equations. BIT19, 79–94 (1979)

    Google Scholar 

  10. Wanner, G.: A short proof on nonlinearA-stability. BIT16, 226–227 (1976)

    Google Scholar 

  11. Wanner, G., Hairer, E., Nørsett, S.P.: Order stars and stability theorems. BIT18, 475–489 (1978)

    Google Scholar 

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Authors and Affiliations

  1. Institute of Numerical Mathematics NTH, N-7034, Trondheim, Norway

    S. P. Nørsett

  2. Section de Mathématiques, Université de Genève, Case postale 124, CH-1211, Genève 24, Switzerland

    G. Wanner

Authors
  1. S. P. Nørsett
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  2. G. Wanner
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Nørsett, S.P., Wanner, G. Perturbed collocation and Runge-Kutta methods. Numer. Math. 38, 193–208 (1981). https://doi.org/10.1007/BF01397089

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  • DOI: https://doi.org/10.1007/BF01397089

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