Theodore E. Simos

Phase-fitted RungeKutta pairs of orders 8(7)

A new phase fitted RungeKutta pair of orders 8(7) which is a modification of a well known explicit RungeKutta pair for the integration of periodic initial value problems is presented. Numerical experiments show the efficiency of the new pair in a wide range of oscillatory problems.

A fifth algebraic order trigonometrically-fitted modified runge-kutta zonneveld method for the numerical solution of orbital problems

A trigonometrically-fitted Runge-Kutta method for the numerical integration of orbital problems is developed. Theoretical and numerical results obtained for some well-known orbital problems show the efficiency of the method.

A phase-fitted Runge-Kutta-Nyström method for the numerical solution of initial value problems with oscillating solutions

Abstract A new Runge–Kutta–Nystrom 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, El-Mikkawy and Prince Runge–Kutta–Nystrom method of algebraic order four with four (three effective) stages. Numerical illustrations indicate that the new method is much more efficient than other methods derived, based on the idea of minimal phase lag or of phase lag of order infinity.

A modified phase-fitted and amplification-fitted Runge-Kutta-Nyström method for the numerical solution of the radial Schrödinger equation

A new Runge-Kutta-Nyström method, with phase-lag and amplification error of order infinity, for the numerical solution of the Schrödinger equation is developed in this paper. The new method is based on the Runge-Kutta-Nyström method with fourth algebraic order, developed by Dormand, El-Mikkawy and Prince. Numerical illustrations indicate that the new method is much more efficient than other methods derived for the same purpose.

Trigonometrically fitted fifth-order runge-kutta methods for the numerical solution of the schrödinger equation

Two trigonometrically fitted methods based on a classical Runge-Kutta method of Kutta-Nystrom are being constructed. The new methods maintain the fifth algebraic order of the classical one but also have some other significant properties. The most important one is that in the local truncation error of the new methods the powers of the energy are lower and that keeps the error at lower values, especially at high values of energy. The error analysis justifies the actual results when integrating the radial Schrodinger equation, where the high efficiency of the new methods is shown.

Trends of the bonding effect on the performance of DFT methods in electric properties calculations: A pattern recognition and metric space approach on some XY2 (X = O, S and Y = H, O, F, S, Cl) molecules

A test set of 10 molecules (open and ring forms of ozone and sulfur dioxide as well as water and hydrogen sulfide and their respective fluoro‐ and chloro‐substituted analogs) of specific atmospheric interest has been formed as to assess the performance of various density functional theory methods in (hyper)polarizability calculations against well‐established ab initio methods. The choice of these molecules was further based on (i) the profound change in the physics between isomeric systems, e.g., open (C2v) and ring (D3h) forms of ozone, (ii) the relation between isomeric forms, e.g., open and ring form of sulfur dioxide (both of C2v symmetry), and (iii) the effect of the substitution, e.g., in fluoro‐ and chloro‐substituted water analogs. The analysis is aided by arguments chosen from the information theory, graph theory, and pattern recognition fields of Mathematics: In brief, a multidimensional space is formed by the methods which are playing the role of vectors with the independent components of the electric properties to act as the coordinates of these vectors, hence the relation between different vectors (e.g., methods) can be quantified by a proximity measure. Results are in agreement with previous studies revealing the acceptable and consistent behavior of the mPW1PW91, B3P86, and PBE0 methods. It is worth noting the remarkable good performance of the double hybrid functionals (namely: B2PLYP and mPW2PLYP) which are for the first time used in calculations of electric response properties.

P-stability, Trigonometric-fitting and the numerical solution of the radial Schrödinger equation

In this paper we will study the importance of the properties of P-stability and Trigonometric-fitting for the numerical integration of the one-dimensional Schrodinger equation. This will be done via the error analysis and the application of the studied methods to the numerical solution of the radial Schrodinger equation.

A new family of exponentially fitted methods

An exponentially-fitted method is developed in this paper. This is a higher-order extension of the dissipative (i.e., nonsymmetric) two-step method first described by Simos and Williams in [1], for the numerical integration of the Schrodinger equation. An application to the bound-states problem and the resonance problem of the radial Schrodinger equation indicates that the new method is more efficient than the classical dissipative method and other well-known methods. Based on the new method and the method of Raptis and Allison [2] a new variable-step method is obtained. The application of the new variable-step method to some coupled differential equations arising from the Schrodinger equation indicates the efficiency of the new approach.

A new trigonometrically-fitted sixth algebraic order P-C algorithm for the numerical solution of the radial schrödinger equation

The new scheme developed in this paper is a new trigonometrically-fitted predictor-corrector (P-C) scheme based on the Adams-Bashforth-Moulton P-C methods. In particular, our predictor is based on the fifth algebraic order Adams-Bashforth scheme and our corrector on the sixth algebraic order Adams-Moulton scheme. We compared the efficiency of this new scheme against well known methods and the numerical experimentations showed that our scheme is more efficient, even when compared to methods that have been specially designed for the solution of the radial Schrodinger equation.

A Zero Dispersion RKN Method for the Numerical Integration of Initial Value Problems with Oscillating Solutions

We will develop a new zero dispersion Runge‐Kutta‐Nystrom method, for the integration of second‐order periodic initial‐value problems. The new method is based on the Runge‐Kutta‐Nystrom method with fourth algebraic order, developed by Dormand, El‐Mikkawy and Prince.