Two-step diagonally-implicit collocation-based methods for Volterra Integral Equations

It is the purpose of this talk to introduce a family of diagonally–implicit continuous methods for the numerical integration of Volterra Integral Equations, which are obtained as a modification of the class of two-step collocation methods introduced in [1, 2]. The derived methods result from the choice of suitable conditions to be satisfied by the collocation polynomial and from the application of special quadrature formulae for the approximation of the increment term, in such a way that the coefficient matrix of the nonlinear system for the computation of the internal stages is lower triangular. The constructed methods have an high uniform order of convergence together with strong stability properties (e.g A-stability). Moreover, the structured shape of the coefficient matrix can be suitably exploited in order to get an efficient implementation in a parallel environment. We present the constructive technique, discuss the order of convergence and provide numerical experiments that confirm the theoretical expectations. References [1] Conte, D., Jackiewicz, Z., and Paternoster, B., Two-step almost collocation methods for Volterra integral equations, Appl. Math. Comput. 204 (2008), 839–853. [2] Conte, D. and Paternoster, B., Multistep collocation methods for Volterra Integral Equations, Appl. Numer. Math. 59 (2009), 1721–1736.