In partial fulfillment of the requirements for a degree of Doctor of Philosophy from Michigan State University.
The method of transfer maps is a powerful tool for the study of weakly nonlinear dynamical systems, especially for the case of large phase space acceptances as in modern particle spectrographs, and the intricate dynamical behaviour of repetitive systems as in circular accelerators and storage rings. The Differential Algebraic (DA) techniques have proven fruitful for various computational problems in beam physics, including the determination of high order Taylor transfer maps.
A new approach, the Remainder-enhanced Differential Algebraic (RDA) method, is presented, which extends the method to allow the determination of remainder bounds for functional dependencies and solutions of ODEs. First, the basic theory of the method is developed and applied to problems of verified optimization and quadrature. Next, schemes are derived that allow the construction of numerical integrators of arbitrary order with rigorous verification of the error for both the integration of individual initial conditions as well as Taylor transfer maps. The methods are based on a differential algebraic fixed point problem which is studied using Schauder's theorem and other functional analysis tools. Employing various compactness arguments on suitable function spaces in combination with the RDA tools, in each integration step a proof of existence of a solution within a tight inclusion is performed.
The resulting computational tools are implemented in the arbitrary order beam physics code COSY INFINITY, and their behavior and performance is studied. Using integration orders around ten and suitable step sizes, rigorous remainder bounds in the range of 10-10 for transfer maps of orders around ten are obtained.
K. Makino (1998)
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