S.N. Chow and E.S. Van Vleck,

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

(1994) SIAM J. Sci. Comp. 15 pp. 959-976.

ABSTRACT

We 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 we 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 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.