A Shadowing Lemma Approach to Global Error Analysis for Initial Value ODEs

The authors show that for dynamical systems that possess a type of piecewise hyperbolicity in which there is no decrease in the number of stable modes, the global error in a numerical approximation may be obtained as a reasonable magnification of the local error. In particular, under certain conditions the authors prove the existence of a trajectory on an infinite time interval of the given ordinary differential equation uniformly close to a given numerically computed orbit of the same differential equation by allowing for different initial conditions. For finite time intervals a general result is proved for obtaining a posteriori bounds on the global error based on computable quantities and on finding and bounding the norm of a right inverse of a particular matrix. Two methods for finding and bounding/estimating the norm of a right inverse are considered. One method is based upon the choice of the pseudo or generalized inverse. The other method is based upon solving multipoint boundary value problems (BV...