Unit:
Τμήμα ΓενικόLibrary of the School of Agricultural Development, Nutrition and Sustainability
Author:
STASINOS PANAGIOTIS
Dissertation committee:
Χαράλαμπος Τσίτουρας , Καθηγητής , Γενικό Τμήμα , ΕΚΠΑ , Επιβλέπων
Θεόδωρος Η. Σίμος , Καθηγητής , Ulyanovsk State Technical University
Ιωάννης Φαμέλης , Καθηγητής , Τμήμα Ηλεκτρολόγων και Ηλεκτρονικών Μηχανικών , ΠΑΔΑ
Ζαχαρούλα Καλογηράτου , Καθηγήτρια , Τμήμα Πληροφορικής , Παν. Δυτικής Μακεδονίας
Φώτιος Κουμπουλής , Καθηγητής , Τμήμα Τεχνολογιών Ψηφιακής Βιομηχανίας , ΕΚΠΑ
Χρήστος Μασούρος , Καθηγητής , Γενικό Τμήμα , ΕΚΠΑ
Βασίλειος Κατσίκης , Αναπληρωτής Καθηγητής , Τμήμα Οικονομικών Επιστημών , ΕΚΠΑ
Original Title:
ΑΡΙΘΜΗΤΙΚΗ ΕΠΙΛΥΣΗ ΔΙΑΦΟΡΙΚΩΝ ΕΞΙΣΩΣΕΩΝ ΚΑΙ ΠΡΟΣ ΤΑ ΠΙΣΩ ΑΝΑΛΥΣΗ ΣΦΑΛΜΑΤΟΣ
Translated title:
NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS AND BACKWARD ERROR ANALYSIS
Summary:
Many problems of this type appear in orbital mechanics they have as a common feature the fact that usually , there is only interest in obtaining the values of the dependent variable , forgetting the values of the derivative . Generally , the most effective way to solve this problem consist in using an initial or starting method and after that , integrating the problem. This is done by means
of a direct integration multi step method.
Methods of this type are the classical Störmer-Cowell formulae but , it has been observed in practice that , when more than two steps are used , the numerical solution spirals inwards. Stefiel and Bettis refer to this phenomenon as orbital unstability. When problems with periodic solution , are integrated numerically , it is desirable that the numerical solution is also periodic , with similar periodic as the analytic one. An appropriate requirement for the numerical methods which integrate periodic problems is P-stability in the sense given by Lambert and Watson. Thus we can obtain multi step linear methods with good periodicity properties for the numerical integration of periodic problems.
In any problem with a periodic solution, even if the frequency of the problem is initially unknown, we have methods with constant coefficients. The methods in this class must be P-stable and this is necessary in the case of problems with extremely oscillating solutions. An important contribution is the work of Hairer in which lower order P-stabe methods have been developed.
Various methods have been proposed in order to overcome the drawback of orbital instability such are the modified methods Störmer-Cowell all of which methods , however , require an a priori knowledge of the frequency. Therefore we will encounter initial value problems whose frequency is known a priori, as well as problems for which we have no knowledge of their frequency. Lambert and Watson dealt with problems for which we have no knowledge of their frequency and defined the conditions under which a linear mutlistep method has a non-vanishing interval of periodicity.
More specifically, Lambert and Watson presented certain linear mutlistep methods of an arbitrary number of steps , which have the property of periodicity when the number of steps as well as the angular frequency move within a defined interval , the interval of periodicity.
The aim of the present doctoral dissertation is the development of faster and more reliable algorithms for the solution of the Schrödinger equation as well as related problems. The results of the research we conducted is the development of such methods which refer to common differential equations with oscillatory of periodic solutions. The reason for their effectiveness , as proven by the analysis we performed, is that in these new methods the phase-lag as well as its derivatives vanish. Another reason is that the methods we developed are of a higher algebraic order.
Main subject category:
Science
Keywords:
backward error analysis , phase-lag , corrector , predictor , Schrodinger Equation
Number of references:
325
