Zacharias A. Anastassi

Construction of an optimized explicit Runge-Kutta-Nyström method for the numerical solution of oscillatory initial value problems

An explicit optimized Runge-Kutta-Nystrom method with four stages and fifth algebraic order is developed. The produced method has variable coefficients with zero phase-lag, zero amplification factor and zero first derivative of the amplification factor. We provide an analysis of the local truncation error of the new method. We also measure the efficiency of the new method in comparison to other numerical methods through the integration of the two-body problem with various eccentricities and three other periodical/oscillatory initial value problems.

A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems

In this article, we develop an explicit symmetric linear phase-fitted four-step method with a free coefficient as parameter. The parameter is used for the optimization of the method in order to solve efficiently the Schrodinger equation and related oscillatory problems. We evaluate the local truncation error and the interval of periodicity as functions of the parameter. We reveal a direct relationship between the periodicity interval and the local truncation error. We also measure the efficiency of the new method for a wide range of possible values of the parameter and compare it to other well known methods from the literature. The analysis and the numerical results help us to determine the optimal values of the parameter, which render the new method highly efficient.

A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems

Abstract In this article we develop a family of three explicit symmetric linear four-step methods. The new methods, with nullified phase-lag, are optimized for the efficient solution of the Schrodinger equation and related oscillatory problems. We perform an analysis of the local truncation error of the methods for the general case and for the special case of the Schrodinger equation, where we show the decrease of the maximum power of the energy in relation to the corresponding classical methods. We also perform a periodicity analysis, where we find that there is a direct relationship between the periodicity intervals of the methods and their local truncation errors. In addition we determine their periodicity regions. We finally compare the new methods to the corresponding classical ones and other known methods from the literature, where we show the high efficiency of the new methods.

A New Symmetric Eight-Step Predictor-Corrector Method For The Numerical Solution Of The Radial Schrödinger Equation And Related Orbital Problems

A new general multistep predictor-corrector (PC) pair form is introduced for the numerical integration of second-order initial-value problems. Using this form, a new symmetric eight-step predictor-corrector method with minimal phase-lag and algebraic order ten is also constructed. The new method is based on the multistep symmetric method of Quinlan–Tremaine,1 with eight steps and 8th algebraic order and is constructed to solve numerically the radial time-independent Schrodinger equation. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with minimal phase-lag is the most efficient of all the compared methods and for all the problems solved.

A symmetric eight-step predictor-corrector method for the numerical solution of the radial Schrödinger equation and related IVPs with oscillating solutions

Abstract In this paper we present a new optimized symmetric eight-step predictor-corrector method with phase-lag of order infinity (phase-fitted). The method is based on the symmetric multistep method of Quinlan–Tremaine, with eight steps and eighth algebraic order and is constructed to solve numerically the radial time-independent Schrodinger equation during the resonance problem with the use of the Woods–Saxon potential. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method 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.

An optimized explicit Runge–Kutta–Nyström method for the numerical solution of orbital and related periodical initial value problems

Abstract In this work a procedure for the construction of an explicit optimized Runge–Kutta–Nystrom method with four stages and fifth algebraic order is provided. The variable coefficients of the preserved method result after nullifying the phase-lag, the dissipative error and the first derivative of the phase-lag. We can see the efficiency of the new method through its local truncation error. Furthermore, we compare the new methodʼs efficiency to other numerical methods. This is shown through the integration of the two-body problem with various eccentricities and of four other initial value problems.