Canonical integration methods for Hamiltonian dynamical systems
Okunbor, Daniel Irowa
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https://hdl.handle.net/2142/21597
Description
Title
Canonical integration methods for Hamiltonian dynamical systems
Author(s)
Okunbor, Daniel Irowa
Issue Date
1993
Doctoral Committee Chair(s)
Skeel, Robert D.
Department of Study
Computer Science
Discipline
Computer Science
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Computer Science
Language
eng
Abstract
Hamiltonian systems possess dynamics (e.g., preservation of volume in phase space and symplectic structure) that call for special numerical integrators, namely canonical methods. Recent research in this aspect have shown that canonical numerical integrators may be needed for Hamiltonian systems. In this thesis, we explore possibilities for the construction of integrators that preserve the property of being canonical. We characterize explicit methods as compositions of 1-stage methods. We find coefficients for methods having minimum work per step. We study in more detail the relationship between canonical, symmetric and order conditions. We also analyze linearized stability. We perform numerical experiments to investigate the importance of being canonical. Examining the solution orbits we discover that numerical integrators possess the same characteristics as the flow of the original system. However, for a variety of test problems we demonstrate equally good results for integrations that merely have the right linearized stability property. We also perform numerical experiments to investigate the improvement that higher-order integrators have over lower-order integrators. This seems to be an interesting research problem, since lower-order methods are currently widely in use in real practical simulations. To direct interest to the application of higher-order methods, serious numerical experiments have to be made to find out their efficiency over lower-order methods. We show that for the two-body problem, the error growth in time for variable step size canonical methods is not in general quadratic. The error growth in time in the case of second-order integrators, canonical or non-canonical turns out to be approximately linear.
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