Picard iteration-based variable-order integrator with dense output employing algorithmic differentiation
Picard iterations are usually employed as a theoretical tool to prove the existence and uniqueness of solutions of initial value problems defined by ordinary differential equations. In this paper, it is shown that they can also be used to construct efficient numerical integration schemes in combination with automatic differentiation techniques. As a matter of fact, they provide truncated Taylor series expansions of the exact solution, so that in this sense, the numerical integrators thus obtained are equivalent to Taylor series methods, although their implementation is different, as well as the required memory resources. \par Two different implementation of Picard iteration-based integrators are presented and applied to the differential equations arising when considering a system of $n$ particles interacting with Coulomb force. The technique allows one, just as in the case of Taylor series methods, to use both variable step-size and order along the integration, as well as endowing the procedure with a dense output, although these issues are not fully explored in the present paper.
scientific article; zbMATH DE number 7015330