Natural continuous extensions of RungeKutta methods for Volterra integral equations of the second kind and their applications
Authors:
A. Bellen, Z. Jackiewicz, R. Vermiglio and M. Zennaro
Journal:
Math. Comp. 52 (1989), 4963
MSC:
Primary 65R20; Secondary 45L05
MathSciNet review:
971402
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Abstract: We consider a very general class of RungeKutta methods for the numerical solution of Volterra integral equations of the second kind, which includes as special cases all the more important methods which have been considered in the literature. The main purpose of this paper is to define and prove the existence of the Natural Continuous Extensions (NCE's) of RungeKutta methods, i.e., piecewise polynomial functions which extend the approximation at the grid points to the whole interval of integration. The particular properties required of the NCE's allow us to construct the tail approximations, which are quite efficient in terms of kernel evaluations.
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 [1]
 C. T. H. Baker & M. S. Keech, "Stability regions in the numerical treatment of Volterra integral equations," SIAM J. Numer. Anal., v. 15, 1978, pp. 394417. MR 0502101 (58:19265)
 [2]
 A. Bellen, "Constrained mesh methods for functional differential equations," in Delay Equations, Approximation and Application (G. Meinardus & G. Nürnberger, eds.), Internat. Ser. Numer. Math., vol. 74, Birkhäuser Verlag, Basel, 1985, pp. 5270. MR 899088 (88f:65226)
 [3]
 A. Bellen, Z. Jackiewicz, R. Vermiglio & M. Zennaro, Natural Continuous Extensions of RungeKutta Methods for Volterra Integral Equations of the Second Kind and Their Applications, Report 65R207, University of Arkansas, Fayetteville, 1987.
 [4]
 A. Bellen, Z. Jackiewicz, R. Vermiglio & M. Zennaro, Stability Analysis of RungeKutta Methods for Volterra Integral Equations of Convolution Type, Report 107, Dept. of Math., Arizona State University, Tempe, 1988.
 [5]
 A. Bellen & M. Zennaro, "Stability properties of interpolants for RungeKutta methods," SIAM J. Numer. Anal., v. 25, 1988, pp. 411432. MR 933733 (89e:65067)
 [6]
 B. A. Bel'tyukov, "An analog of the RungeKutta method for solution of nonlinear Volterra type integral equations," Differential Equations, v. 1, 1965, pp. 417426.
 [7]
 H. Brunner, "Collocation methods for onedimensional Fredholm and Volterra integral equations," in The State of the Art in Numerical Analysis (A. Iserles & M. J. D. Powell, eds.), Clarendon Press, Oxford, 1987, pp. 563600. MR 921678 (89m:65112)
 [8]
 H. Brunner, E. Hairer & S. P. Nørsett, "RungeKutta theory for Volterra integral equations of the second kind," Math. Comp., v. 39, 1982, pp. 147163. MR 658219 (83f:65203)
 [9]
 H. Brunner & S. P. Nørsett, "Superconvergence of collocation methods for Volterra and Abel integral equations of the second kind," Numer. Math., v. 36, 1981, pp. 347358. MR 614853 (83e:65202)
 [10]
 H. Brunner & P. J. van der Houwen, The Numerical Solution of Volterra Equations, NorthHolland, Amsterdam, 1986. MR 871871 (88g:65136)
 [11]
 J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations: RungeKutta and General Linear Methods, Wiley, Chichester, New York, 1987. MR 878564 (88d:65002)
 [12]
 E. Hairer, "Order conditions for numerical methods for partitioned ordinary differential equations," Numer. Math., v. 36, 1981, pp. 431445. MR 614858 (82j:65047)
 [13]
 E. Hairer & C. Lubich, "On the stability of VolterraRungeKutta methods," SIAM J. Numer. Anal., v. 21, 1984, pp. 123135. MR 731217 (85m:65134)
 [14]
 E. Hairer, C. Lubich & S. P. Nørsett, "Order of convergence of onestep methods for Volterra integral equations of the second kind," SIAM J. Numer. Anal., v. 20, 1983, pp. 569579. MR 701097 (84g:65163)
 [15]
 S. P. Nørsett & G. Wanner, "The realpole sandwich for rational approximations and oscillation equations," BIT, v. 19, 1979, pp. 7994. MR 530118 (81d:65040)
 [16]
 P. Pouzet, "Etude en vue de leur traitement numérique des équations intégrales de type Volterra," Rev. Française Traitement Information (Chiffres), v. 6, 1963, pp. 79112. MR 0152152 (27:2132)
 [17]
 P. J. van der Houwen, "Convergence and stability results in RungeKutta type methods for Volterra integral equations of the second kind," BIT, v. 20, 1980, pp. 375377. MR 595219 (82f:65139)
 [18]
 P. J. van der Houwen, P. H. M. Wolkenfelt & C. T. H. Baker, "Convergence and stability analysis for modified RungeKutta methods in the numerical treatment of secondkind Volterra integral equations," IMA J. Numer. Anal., v. 1, 1981, pp. 303328. MR 641312 (83a:65127)
 [19]
 M. Zennaro, "Natural continuous extensions of RungeKutta methods," Math. Comp., v. 46, 1986, pp. 119133. MR 815835 (86m:65083)
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DOI:
http://dx.doi.org/10.1090/S00255718198909714023
PII:
S 00255718(1989)09714023
Article copyright:
© Copyright 1989
American Mathematical Society
