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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 49-63 |
| Number of pages | 15 |
| Journal | Mathematics of Computation |
| Volume | 52 |
| Issue number | 185 |
| DOIs | |
| State | Published - 1989 |
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ASJC Scopus subject areas
- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics
Cite this
First occurrence prime gaps. / Bellen, A.; Jackiewicz, Zdzislaw; Vermiglio, R.; Zennaro, M.
In: Mathematics of Computation, Vol. 52, No. 185, 1989, p. 49-63.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - First occurrence prime gaps
AU - Bellen, A.
AU - Jackiewicz, Zdzislaw
AU - Vermiglio, R.
AU - Zennaro, M.
PY - 1989
Y1 - 1989
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84966230115&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84966230115&partnerID=8YFLogxK
U2 - 10.1090/S0025-5718-1989-0971402-3
DO - 10.1090/S0025-5718-1989-0971402-3
M3 - Article
AN - SCOPUS:84966230115
VL - 52
SP - 49
EP - 63
JO - Mathematics of Computation
JF - Mathematics of Computation
SN - 0025-5718
IS - 185
ER -