Stepsize selection in the rigorous defect control of Taylor series methods

Modern numerical methods for initial-value problems (IVPs) in ordinary differential equations (ODE) produce a (piecewise) differentiable numerical solution. The defect or residual evaluated at it induces a perturbed ODE that is satisfied exactly by this solution. Defect control methods try to ensure that a norm of the defect is bounded by a user-specified tolerance. This paper justifies and implements a simple and effective stepsize selection strategy and an overall method for both validated and non-validated defect control of an explicit Taylor series method for IVP ODEs. In validated mode, this method guarantees that the defect is bounded by a user-specified tolerance.