Phase-fitted Discrete Lagrangian Integrators

Phase fitting has been extensively used during the last years to improve the behaviour of numerical integrators on oscillatory problems. In this work, the benefits of the phase fitting technique are embedded in discrete Lagrangian integrators. The results show improved accuracy and total energy behaviour in Hamiltonian systems. Numerical tests on the long term integration (100000 periods) of the 2-body problem with eccentricity even up to 0.95 show the efficiency of the proposed approach. Finally, based on a geometrical evaluation of the frequency of the problem, a new technique for adaptive error control is presented

New modified Runge–Kutta–Nyström methods for the numerical integration of the Schrödinger equation

AbstractIn this work we construct new Runge–Kutta–Nyström methods especially designed to integrate exactly the test equation y″=−w2y. We modify two existing methods: the Runge–Kutta–Nyström methods of fifth and sixth order. We apply the new methods to the computation of the eigenvalues of the Schrödinger equation with different potentials such as the harmonic oscillator, the doubly anharmonic oscillator and the exponential potential

Symplectic integrators for the numerical solution of problems with oscillatory behavior of the solution

This thesis deals with the numerical solution of the Hamiltonian problems by using symplectic integrators. More specifically, explicit symplectic methods are examined and constructed, in case the Hamiltonian is separable. Initially, two new symplectic methods, third and fourth order, are constructed and implemented for the solution of the time independent Schrodinger equation. The Schrodinger equation is first transformed into a Hamiltonian canonical system. Moreover, almost all the existing symplectic methods, up to the eighth order, are also implemented, in order to compare, in general terms, their effectiveness. Furthermore, a method, which is based on the third order symplectic integrator, is used for the numerical solution of the two dimensional problem of the Schrodinger’s equation through the partial discretization. In addition, exponentially and trigonometrically fitted symplectic methods are implemented for the numerical solution of the general Hamiltonian problem, as well as for the computation of eigenvalues of the Schrodinger equation.Στην παρούσα διδακτορική διατριβή μελετάται η αριθμητική επίλυση των Χαμιλτονιανών προβλημάτων, με την εφαρμογή συμπλεκτικών ολοκληρωτών. Ειδικότερα μελετώνται και κατασκευάζονται έμμεσες συμπλεκτικές μέθοδοι για την περίπτωση που η Χαμιλτονιανή είναι χωριζόμενων μεταβλητών. Αρχικά δημιουργούνται δύο νέες συμπλεκτικές μέθοδοι, τρίτης και τέταρτης τάξης οι οποίες και εφαρμόζονται στην επίλυση της χρονικά ανεξάρτητης εξίσωσης του Schrodinger, αφού πρώτα αυτή μετατραπεί σε ένα κανονικό Χαμιλτονιανό σύστημα. Εφαρμόζονται επίσης και όλες σχεδόν οι υπάρχουσες συμπλεκτικές μέθοδοι, μέχρι και όγδοης τάξης, για μια γενικότερη σύγκριση της αποτελεσματικότητας τους. Κατόπιν κατασκευάζεται μέθοδος που στηρίζεται σε συμπλεκτικό ολοκληρωτή τρίτης τάξης, για την αριθμητική επίλυση της διδιάστατης εξίσωσης του Schrodinger με την τεχνική της ημιδιακριτοποίησης. Στην συνέχεια αναπτύσσονται εκθετικά και τριγωνομετρικά προσαρμοσμένες συμπλεκτικές μέθοδοι, για την αριθμητική επίλυση του γενικού Χαμιλτονιανού προβλήματος, καθώς και για την εύρεση ιδιοτιμών στην εξίσωση του Schrodinger

High Order Two-Derivative Runge-Kutta Methods with Optimized Dispersion and Dissipation Error

In this work we consider explicit Two-derivative Runge-Kutta methods of a specific type where the function f is evaluated only once at each step. New 7th order methods are presented with minimized dispersion and dissipation error. These are two methods with constant coefficients with 5 and 6 stages. Also, a modified phase-fitted, amplification-fitted method with frequency dependent coefficients and 5 stages is constructed based on the 7th order method of Chan and Tsai. The new methods are applied to 4 well known oscillatory problems and their performance is compared with the methods in that of Chan and Tsai.The numerical experiments show the efficiency of the derived methods

Exponentially fitted symplectic Runge-Kutta-Nyström methods

In this work we consider symplectic Runge Kutta Nystr¨om (SRKN) methods with three stages. We construct a fourth order SRKN with constant coefficients and a trigonometrically fitted SRKN method. We apply the new methods on the two-dimentional harmonic oscillator, the Stiefel-Bettis problem and on the computation of the eigenvalues of the Schr¨odinger equation

Symplectic Runge-Kutta-Nystr?m Methods with Phase-Lag Oder 8 and Infinity

In this work we consider Symplectic Runge Kutta Nystr¨om methods with five stages. A new fourth algebraic order method with phase-lag order eight is presented. Also the symplectic Runge Kutta Nystr¨om of Calvo and Sanz Serna with five stages and fourth order is modified to produce a phase-fitted method. We apply the new methods on several Hamiltonian systems and on the computation of the eigenvalues of the Schr¨odinger Equation

Modeling Regional Employment. An Application in High Technology Sectors in Greece

AbstractMathematical models of competing species defined by ordinary differential equations are considered to construct a model for analysing the spatial dimension of employment. Analysis of the points of equilibrium is given in an application of the model using data on employment in High Technology Sectors (HTS), for the period 1999-2008, for Greece. The findings suggest that there is tendency of movement in employees in HTS in Greece from the Attica region to the Rest of Greece. In the long-run equilibrium it was found that the employment in Attica will be between the levels of 2007 and 2008 while it will increase in the periphery of Greece

Computational method for approximating the behaviour of a triopoly: An application to the mobile telecommunications sector in Greece

Computational biology models of the Volterra-Lotka family, known as competing species models, are used for modelling a triopoly market, with application to the mobile telecommunications in Greece. Using a data sample for 1999–2016, parameter estimation with nonlinear least squares is performed. The findings show that the proportional change in the market share of the two largest companies, Cosmote and Vodafone, depends negatively on the market share of each other. Further, the market share of the marker leader, Cosmote, depends positively on the market share of the smallest company, Wind. The proportional change in the market share of Wind, depends negatively on the market share of the largest company Cosmote but it depends positively by the change in the market share by the second company, Vodafone. In the long-run it was found that the market shares tend to the stable equilibrium point where all three companies will survive with Cosmote having a projected number after eleven years (in 2030) of approximately 7.3 million subscribers, Vodafone 4.9 and Wind 3.7, the total number of projected market size being approximately 16 million customers. Copyright © 2021 Inderscience Enterprises Ltd