Validated B-series and Runge-Kutta pairs

Considering validated Runge-Kutta schemes requires the computation of local truncation error. Computation of such error is expensive in term of complexity, thus few techniques have been proposed to compute or at least estimate it. For example, Runge-Kutta pairs provide an error estimation by computing the difference of two schemes of different orders while sharing the same stages. With the goal of proposing a validated scheme, an approximation is not sufficient, but could be helpful after some adjustments. In this paper, we compute a new pair and show its performance on a non-trivial problem. Two new techniques for the local truncation error computation are then presented: a symbolic based approach dedicated to a specific problem and a preliminary study of pair’s approximation.