A new phase-fitted eight-step symmetric embedded predictor–corrector method (EPCM) for orbital problems and related IVPs with oscillating solutions

Our new phase-fitted embedded predictor–corrector method (EPCM) presented here is based on the multistep symmetric method of Quinlan–Tremaine (1990), with eight steps and eighth algebraic order and constructed to solve numerically the two-dimensional Kepler problem. It can also be used to integrate other orbital problems and related IVPs with oscillatory solutions. First we present a EPCM (Panopoulos et al. (2011) and Panopoulos et al. (2013)) pair form. From this form we construct a new eight-step method. The new scheme has algebraic order ten and infinite order of phase-lag. We tested the efficiency of our newly developed scheme against to some recently constructed optimized methods and other well known methods from the literature. We measure the efficiency of the methods and conclude that the new scheme is noticeably more efficient of all the compared methods and for all the problems solved, including the radial Schrödinger equation.     

Runge-Kutta-Nyström interpolants for the numerical integration of special second-order periodic initial-value problems

The theory of the phase-lag analysis for Runge-Kutta-Nyström methods and Runge-Kutta-Nyström interpolants is developed in this paper. Also a new Runge-Kutta-Nyström method with interpolation properties is developed to integrate second-order differential equations of the formu″(t)=f(t,u) when they possess an oscillatory solution.     

Symplectic Partitioned Runge-Kutta methods with minimal phase-lag

Symplectic Partitioned Runge-Kutta (SPRK) methods with minimal phase-lag are derived. Specifically two new symplectic methods are constructed of second and third order with fifth phase-lag order. The methods are tested on the numerical integration of Hamiltonian problems and the Schrödinger equation.     

A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

Many simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge-Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic order     

关于求解二阶周期性初值问题具有极小相位延迟的P—稳定的Hybrid方法

本文提出了一类求解二阶周期性初值问题y"=f(x,y)的具有极小相位延迟的P—稳定的Hybrid方法。其精度与相位延迟的阶均比[1～3]中的同类方法高。本文10阶方法M_(10)(β)只须求f的二阶全导数,6阶方法M_6(β)及8阶方法M_8(β,γ)不需求f的全导数。数值结果表明,本文所构造的方法是有效的。     

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

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 Schrödinger 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.     

A generator of dissipative methods for the numerical solution of the Schrödinger equation

In this paper a generator of hybrid methods with minimal phase-lag is developed for the numerical solution of the Schrödinger equation and related problems. The generator's methods are dissipative and are of eighth algebraic order. In order to have minimal phase-lag with the new methods, their coefficients are determined automatically. Numerical results obtained by their application to some well known problems with periodic or oscillating solutions and to the coupled differential equations of the Schrödinger type indicate the efficiency of these new methods.     

An explicit eighth order method with minimal phase-lag for the numerical solution of the Schrödinger equation

A new explicit hybrid eighth algebraic order two-step method with phase-lag of order twelve is developed for computing eigenvalues and resonances of the one-dimensional Schrödinger equation. Based on this new method and on the method developed recently by Simos we obtain a new variable-step procedure for the numerical integration of the Schrödinger equation. Numerical results obtained for the integration of the resonance problem for the well known case of the Woods-Saxon potential and for the integration of the eigenvalue problem for the well known case of the Morse potential show that this new method is better than other variable-step methods.