On two-step continuous methods for Volterra Integral Equations

The aim of this talk is to present highly stable collocation based numerical methods for Volterra Integral Equations (VIEs). As it is well known, a collocation method is based on the idea of approximating the exact solution of a given integral equation with a suitable function belonging to a chosen nite dimensional space, usually a piecewise algebraic polynomial, which satises the integral equation exactly on a certain subset of the integration interval (called the set of collocation points). The collocation technique allows the derivation of methods having many desirable properties. In fact, collocation methods provide an approximation at each point of the integration interval to the solution of the equation, thus leading, in general, to a cheap variable stepsize implementation. Moreover, the collocation function can be expressed as a linear combination of functions ad hoc for the problem we are integrating, in order to better reproduce the qualitative behaviour of the solution. Two-step collocation and modied collocation methods have already been developed in the context of Ordinary Dierential Equations [6,7] and VIEs [3{5] with the aim of increasing the order of classical one-step methods, without any increase of the computational cost, maintaining desirable stability properties. In particular we aim to present new results obtained in the context of two-step almost collocation methods for VIEs, i.e. methods obtained by relaxing some collocation and/or interpolation conditions in order to obtain high uniform order of convergence together with strong stability properties. We will exploit the continuous order conditions in order to provide a possible error estimation which will be at the basis of a variable stepsize implementation and show numerical experiments that conrm the theoretical expectations. References: [1] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, 2004. [2] H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, CWI Monographs, 3, North-Holland, Amsterdam, 1986. [3] Conte, D., D'Ambrosio, and Paternoster, B., Two-step diagonally-implicit collocation-based methods for Volterra Integral Equations, Appl. Numer. Math. 62, 1312-1324 (2012). [4] Conte, D., Jackiewicz, Z., and Paternoster, B., Two-step almost collocation methods for Volterra integral equations, Appl. Math. Comput. 204, 839{853 (2008). [5] Conte, D. and Paternoster, B., Multistep collocation methods for Volterra Integral Equations, Appl. Numer. Math. 59, 1721{1736 (2009). [6] R. D'Ambrosio, M. Ferro, Z. Jackiewicz, B.Paternoster, Two-step almost collocation methods for Ordinary Dierential Equations, Numerical Algorithms 53(2-3), 195{217 (2010). [7] R. D'Ambrosio, Z. Jackiewicz (2011), Construction and implementation of highly stable two-step continuous methods for sti dierential systems, Mathematics and Computers in Simulation 81 (9), 1707-1728 (2011).