Rigorous Integration of Flows and ODEs using Taylor Models
AbstractTaylor models combine the advantages of numerical methods
and algebraic approaches of efficiency, tightly controlled
recourses, and the ability to handle very complex problems
with the advantages of symbolic approaches, in particularly
the ability to be rigorous and to allow the treatment of functional
dependencies instead of merely points. The resulting
differential algebraic calculus involving an algebra with differentiation
and integration is particularly amenable for the
study of ODEs and PDEs based on fixed point problems
from functional analysis. We describe the development of
rigorous tools to determine enclosures of flows of general
nonlinear differential equations based on Picard iterations.
Particular emphasis is placed on the development of methods
that have favorable long term stability, which is achieved
using suitable preconditioning and other methods. Applications
of the methods are presented, including determinations
of rigorous enclosures of flows of ODEs in the theory
of chaotic dynamical systems.
K. Makino, M. Berz,
Symbolic Numeric Computation 2009, (2009) 79-84
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