Natural continuous extensions of Runge-Kutta 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), 49-63

MSC:
Primary 65R20; Secondary 45L05

DOI:
https://doi.org/10.1090/S0025-5718-1989-0971402-3

MathSciNet review:
971402

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a very general class of Runge-Kutta 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 Runge-Kutta 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.

**[1]**Christopher T. H. Baker and Malcolm S. Keech,*Stability regions in the numerical treatment of Volterra integral equations*, SIAM J. Numer. Anal.**15**(1978), no. 2, 394–417. MR**0502101**, https://doi.org/10.1137/0715025**[2]**A. Bellen,*Constrained mesh methods for functional-differential equations*, Delay equations, approximation and application (Mannheim, 1984) Internat. Schriftenreihe Numer. Math., vol. 74, Birkhäuser, Basel, 1985, pp. 52–70. MR**899088****[3]**A. Bellen, Z. Jackiewicz, R. Vermiglio & M. Zennaro,*Natural Continuous Extensions of Runge-Kutta Methods for Volterra Integral Equations of the Second Kind and Their Applications*, Report 65R20-7, University of Arkansas, Fayetteville, 1987.**[4]**A. Bellen, Z. Jackiewicz, R. Vermiglio & M. Zennaro,*Stability Analysis of Runge-Kutta Methods for Volterra Integral Equations of Convolution Type*, Report 107, Dept. of Math., Arizona State University, Tempe, 1988.**[5]**Alfredo Bellen and Marino Zennaro,*Stability properties of interpolants for Runge-Kutta methods*, SIAM J. Numer. Anal.**25**(1988), no. 2, 411–432. MR**933733**, https://doi.org/10.1137/0725028**[6]**B. A. Bel'tyukov, "An analog of the Runge-Kutta method for solution of nonlinear Volterra type integral equations,"*Differential Equations*, v. 1, 1965, pp. 417-426.**[7]**H. Brunner,*Collocation methods for one-dimensional Fredholm and Volterra integral equations*, The state of the art in numerical analysis (Birmingham, 1986) Inst. Math. Appl. Conf. Ser. New Ser., vol. 9, Oxford Univ. Press, New York, 1987, pp. 563–600. MR**921678****[8]**H. Brunner, E. Hairer, and S. P. Nørsett,*Runge-Kutta theory for Volterra integral equations of the second kind*, Math. Comp.**39**(1982), no. 159, 147–163. MR**658219**, https://doi.org/10.1090/S0025-5718-1982-0658219-8**[9]**Hermann Brunner and Syvert P. Nørsett,*Superconvergence of collocation methods for Volterra and Abel integral equations of the second kind*, Numer. Math.**36**(1980/81), no. 4, 347–358. MR**614853**, https://doi.org/10.1007/BF01395951**[10]**H. Brunner and P. J. van der Houwen,*The numerical solution of Volterra equations*, CWI Monographs, vol. 3, North-Holland Publishing Co., Amsterdam, 1986. MR**871871****[11]**J. C. Butcher,*The numerical analysis of ordinary differential equations*, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1987. Runge\mhy Kutta and general linear methods. MR**878564****[12]**E. Hairer,*Order conditions for numerical methods for partitioned ordinary differential equations*, Numer. Math.**36**(1980/81), no. 4, 431–445. MR**614858**, https://doi.org/10.1007/BF01395956**[13]**E. Hairer and Ch. Lubich,*On the stability of Volterra-Runge-Kutta methods*, SIAM J. Numer. Anal.**21**(1984), no. 1, 123–135. MR**731217**, https://doi.org/10.1137/0721008**[14]**E. Hairer, Ch. Lubich, and S. P. Nørsett,*Order of convergence of one-step methods for Volterra integral equations of the second kind*, SIAM J. Numer. Anal.**20**(1983), no. 3, 569–579. MR**701097**, https://doi.org/10.1137/0720037**[15]**S. P. Nørsett and G. Wanner,*The real-pole sandwich for rational approximations and oscillation equations*, BIT**19**(1979), no. 1, 79–94. MR**530118**, https://doi.org/10.1007/BF01931224**[16]**P. Pouzet,*Étude en vue de leur traitement numérique des équations intégrales de type Volterra*, Rev. Franç. Traitement Information Chiffres**6**(1963), 79–112 (French). MR**0152152****[17]**P. J. van der Houwen,*Convergence and stability results in Runge-Kutta type methods for Volterra integral equations of the second kind*, BIT**20**(1980), no. 3, 375–377. MR**595219**, https://doi.org/10.1007/BF01932780**[18]**P. J. van der Houwen, P. H. M. Wolkenfelt, and C. T. H. Baker,*Convergence and stability analysis for modified Runge-Kutta methods in the numerical treatment of second-kind Volterra integral equations*, IMA J. Numer. Anal.**1**(1981), no. 3, 303–328. MR**641312**, https://doi.org/10.1093/imanum/1.3.303**[19]**M. Zennaro,*Natural continuous extensions of Runge-Kutta methods*, Math. Comp.**46**(1986), no. 173, 119–133. MR**815835**, https://doi.org/10.1090/S0025-5718-1986-0815835-1

Retrieve articles in *Mathematics of Computation*
with MSC:
65R20,
45L05

Retrieve articles in all journals with MSC: 65R20, 45L05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0971402-3

Article copyright:
© Copyright 1989
American Mathematical Society