THEMATIC PROGRAMS

Numerical and Computational Challenges in Science and Engineering Program

Wayne Hayes, The Fields Institute

LECTURES

February 15, 2002, 10:00 am

Shadowing Numerical Solutions of ODEs, with Applications

Many dynamical systems studied today are "chaotic", which implies that two solutions whose initial conditions differ by an arbitrarily small amount will diverge exponentially away from each other. This is worrisome to those who perform numerical studies of these systems, because it means that numerical errors cause numerical solutions to diverge exponentially away from the exact solution with the same initial conditions. Surprisingly, for some (but not all) chaotic systems, it turns out that there may exist an exact solution to the differential equation, called a "shadow", that remains near the numerical one, although it is not the exact solution that started with the same initial conditions as the numerical one. If the study of the system does not depend upon the precise choice of initial conditions, then the existence of a shadow is strong evidence that the numerical solution faithfully exhibits the properties of exact solutions to the problem being studied. In this talk, I will outline how shadows of numerical solutions of ODEs can be found, and discuss shadowing of the large gravitational N-body problem.

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