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

Abstract

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 (BVPs) with the choice of boundary conditions motivated by the piecewise hyperbolicity concept. Numerical results are presented for the logistic equation, the forced pendulum equation, and the space discretized Chafee–Infante equation.