Implicit RungeKutta methods of optimal order for Volterra integrodifferential equations
Author:
Hermann Brunner
Journal:
Math. Comp. 42 (1984), 95109
MSC:
Primary 65R20; Secondary 45J05, 45L10
MathSciNet review:
725986
Fulltext PDF Free Access
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Abstract: Implicit RungeKutta methods with m stages and optimal order for the approximate solution of Volterra integrodifferential equations can be viewed as fully discretized collocation methods in certain polynomial spline spaces. The choice of the quadrature formulas needed for the full discretization of the collocations is investigated, and it is shown that, in contrast to ordinary differential equations, there exist (for fixed m) several optimal methods.
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 C. T. H. Baker, "Initial value problems for Volterra integrodifferential equations," in Modern Numerical Methods for Ordinary Differential Equations (G. Hall and J. M. Watt, eds.), Clarendon Press, Oxford, 1976, p. 296307.
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 [5]
 C. W. Cryer, "Numerical methods for functional differential equations," in Delay and Functional Differential Equations and Their Applications (K. Schmitt, ed.), Academic Press, New York, 1972, pp. 17101. MR 0388820 (52:9654)
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 A. Feldstein & J. R. Sopka, "Numerical methods for nonlinear integrodifferential equations," SIAM J. Numer. Anal., v. 11, 1974, pp. 826846. MR 0375816 (51:12006)
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 A. Ghizzetti & A. Ossicini, Quadrature Formulae, Birkhäuser Verlag, Basel, 1970. MR 0269116 (42:4012)
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 A. Guillou & J. L. Soulé, "La résolution numérique des problèmes différentiels aux conditions initiales par des méthodes de collocation," RAIRO Sér. Rouge, v. 3, 1969, pp. 1744. MR 0280008 (43:5729)
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 C. Lubich, RungeKutta Theory for Volterra Integrodifferential Equations, Preprint No. 131, Sonderforschungsbereich 123, University of Heidelberg, 1981.
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 A. Makroglou, "Convergence of a blockbyblock method for nonlinear Volterra integrodifferential equations," Math. Comp., v. 35, 1980, pp. 783796. MR 572856 (81g:65180)
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 W. L. Mocarsky, "Convergence of stepbystep methods for nonlinear Volterra integrodifferential equations," J. Inst. Math. Appl., v. 8, 1971, pp. 235239. MR 0287734 (44:4937)
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 S. P. Nørsett, "Collocation and perturbed collocation methods," in Numerical Analysis, Dundee 1979 (G. A. Watson, ed.), Lecture Notes in Math., Vol. 773, SpringerVerlag, Berlin, Heidelberg, New York, 1980, pp. 119132. MR 569466 (81h:65078)
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 S. P. Nørsett & G. Wanner, "The realpole sandwich for rational approximations and oscillation equations," BIT, v. 19, 1979, pp. 7994. MR 530118 (81d:65040)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198407259866
PII:
S 00255718(1984)07259866
Keywords:
Volterra integrodifferential equations,
collocation methods,
implicit RungeKutta methods of optimal order
Article copyright:
© Copyright 1984
American Mathematical Society
