Utilizing Schauder's fixed point theorem on certain suitable compact and convex sets of functions, we show how explicit n-th order integrators can be developed that provide verified n-th order inclusions of a solution of the ODE. The method can be used not only for the computation of solutions through a single initial condition, but also to establish the functional dependency between initial and final conditions, the so-called flow of the ODE. The latter can be used efficiently for a substantial reduction of the wrapping effect.
Examples of the application of the method to conventional initial value problems as well as flows are given. The orders of the integration range up to twelve, and the verified inclusions of up to thirteen digits of accuracy have been demanded and obtained.
M. Berz, K. Makino, Reliable Computing 4 (1998) 361-369
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