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T. E. Simos

Publications and source records attributed to T. E. Simos.

11 recordsLinked to original sources
arXiv2008

A Family of Runge-Kutta Methods with Zero Phase-Lag and Derivatives for the Numerical Solution of the Schr\"odinger Equation and Related Problems

We construct a family of two new optimized explicit Runge-Kutta methods with zero phase-lag and derivatives for the numerical solution of the time-independent radial Schr\"odinger equation and related ordinary differential equations with oscillating solutions. The numerical results show the superiority of the new technique of nullifying both the phase-lag and its derivatives.

math.NA↗
arXiv2008

Two Optimized Symmetric Eight-Step Implicit Methods for Initial-Value Problems with Oscillating Solutions

In this paper we present two optimized eight-step symmetric implicit methods with phase-lag order ten and infinite (phase-fitted). The methods are constructed to solve numerically the radial time-independent Schr\"odinger equation with the use of the Woods-Saxon potential. They can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the two new methods to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved.

math.NA↗
arXiv2008

High Order Multistep Methods with Improved Phase-Lag Characteristics for the Integration of the Schr\"odinger Equation

In this work we introduce a new family of twelve-step linear multistep methods for the integration of the Schr\"odinger equation. The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both the phase lag function and its first derivatives at a specific frequency. This results in decreasing the sensitivity of the integration method on the estimated frequency of the problem. The efficiency of the new family of methods is proved via error analysis and numerical applications.

math.NA↗
arXiv2008

High Order Phase Fitted Multistep Integrators for the Schr\"odinger Equation with Improved Frequency Tolerance

In this work we introduce a new family of 14-steps linear multistep methods for the integration of the Schr\"odinger equation. The new methods are phase fitted but they are designed in order to improve the frequency tolerance. This is achieved by eliminating the first derivatives of the phase lag function at the fitted frequency forcing the phase lag function to be '\textit{flat}' enough in the neighbor of the fitted frequency. The efficiency of the new family of methods is proved via error analysis and numerical applications.

math.NA↗
arXiv2008

Zero Dispersion and Zero Dissipation Implicit Runge-Kutta Methods for the Numerical Solution of Oscillating IVPs

In this paper we present two new methods based on an implicit Runge-Kutta method Gauss which is of algebraic order fourth and has two stages: the first one has zero dispersion and the second one has zero dispersion and zero dissipation. The efficiency of these methods is measured while integrating the radial Schr\"odinger equation and other well known initial value problems.

math.NA↗
arXiv2008

A Phase-Fitted Runge-Kutta-Nystr\"om method for the Numerical Solution of Initial Value Problems with Oscillating Solutions

A new Runge-Kutta-Nystr\"om method, with phase-lag of order infinity, for the integration of second-order periodic initial-value problems is developed in this paper. The new method is based on the Dormand and Prince Runge-Kutta-Nystr\"om method of algebraic order four\cite{pa}. Numerical illustrations indicate that the new method is much more efficient than the classical one.

math.NA↗
arXiv2008

A Family of Multistep Methods with Zero Phase-Lag and Derivatives for the Numerical Integration of Oscillatory ODEs

In this paper we develop a family of three 8-step methods, optimized for the numerical integration of oscillatory ordinary differential equations. We have nullified the phase-lag of the methods and the first r derivatives, where r=1,2,3. We show that with this new technique, the method gains efficiency with each derivative of the phase-lag nullified. This is the case for the integration of both the Schrodinger equation and the N-body problem. A local truncation error analysis is performed, which, for the case of the Schrodinger equation, also shows the connection of the error and the energy, revealing the importance of the zero phase-lag derivatives. Also the stability analysis shows that the methods with more derivatives vanished, have a bigger interval of periodicity.

math.NA↗
arXiv2008

Phase Lag Sensitivity Analysis for Numerical Integration

In the field of numerical integration, methods specially tuned on oscillating functions, are of great practical importance. Such methods are needed in various branches of natural sciences, particularly in physics, since a lot of physical phenomena exhibit a pronounced oscillatory behavior. Among others, probably the most important tool used to construct efficient methods for oscillatory problems is the exponential (trigonometric) fitting. The basic characteristic of these methods is that their phase lag vanishes at a predefined frequency. In this work, we introduce a new tool which improves the behavior of exponentially fitted numerical methods. The new technique is based on the vanishing of the first derivatives of the phase lag function at the fitted frequency. It is proved in the text that these methods present improved characteristics in oscillatory problems.

math.NA↗
arXiv2008

A New Family of Multistep Methods with Improved Phase Lag Characteristics for the Integration of Orbital Problems

In this work we introduce a new family of ten-step linear multistep methods for the integration of orbital problems. The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both the phase lag function and its first derivatives at a specific frequency. The efficiency of the new family of methods is proved via error analysis and numerical applications.

math.NA↗
arXiv2008

Synchronization in Complex Systems Following the Decision Based Queuing Process: The Rhythmic Applause as a Test Case

Living communities can be considered as complex systems, thus a fertile ground for studies related to their statistics and dynamics. In this study we revisit the case of the rhythmic applause by utilizing the model proposed by Vázquez et al. [A. Vázquez et al., Phys. Rev. E 73, 036127 (2006)] augmented with two contradicted {\it driving forces}, namely: {\it Individuality} and {\it Companionship}. To that extend, after performing computer simulations with a large number of oscillators we propose an explanation on the following open questions (a) why synchronization occurs suddenly, and b) why synchronization is observed when the clapping period ($T_c$) is $1.5 \cdot T_s < T_c < 2.0 \cdot T_s$ ($T_s$ is the mean self period of the spectators) and is lost after a time. Moreover, based on the model, a weak preferential attachment principle is proposed which can produce complex networks obeying power law in the distribution of number edges per node with exponent greater than 3.

cond-mat.stat-mech↗