EFFICIENT SIXTH ORDER P-STABLE METHODS WITH MINIMAL LOCAL TRUNCATION ERROR FOR y"=f(x,y)

1. IntroductionWe consider a class of direct hybrid methods proposed in [11 for solving the second orderinitial value problemy" = f(t,y), y(0),y'(0) given (1.1)The basic method has the formandHere t. = nh and we define t.l.. = t. I aih, i = 1, 2 and n=0,1…     

On energy conservation and P-stability

When applied to the classical dynamical equations, generally, conventional time-stepping methods cannot conserve the total energy exactly, but up to their algebraic order. To obtain an energy-conserving method, we add an energy calibrated parameter to some conventional ones. The periodic stability and phase-lag property of this type of energy-conserving methods are investigated in this note. Some numerical results are reported to illustrate our conclusions.     

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?dinger 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. T. E. Simos—Highly Cited Researcher, Active Member of the European Academy of Sciences and Arts.     

High order multistep methods with improved phase-lag characteristics for the integration of the Schr?dinger equation

In this work we introduce a new family of 12-step linear multistep methods for the integration of the Schr?dinger 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. T. E. Simos is a highly cited researcher, active member of the European Academy of Sciences and Arts. Corresponding member of the European Academy of Sciences, corresponding member of European Academy of Arts, Sciences and Humanities.     

A new methodology for the development of numerical methods for the numerical solution of the Schr?dinger equation

In the present paper we introduce a new methodology for the construction of numerical methods for the approximate solution of the one-dimensional Schr?dinger equation. The new methodology is based on the requirement of vanishing the phase-lag and its derivatives. The efficiency of the new methodology is proved via error analysis and numerical applications.     

A modified Runge-Kutta method for the numerical solution of ODE''s with oscillation solutions

A modified Runge-Kutta method with minimal phase-lag is developed for the numerical solution of Ordinary Differential Equations with oscillating solutions. The method is based on the accurate Runge-Kutta method of Sharp and Smart RK4SS(5) (see [1]) of order five. Numerical and theoretical results show that this new approach is more efficient, compared with the fifth order Runge-Kutta Sharp and Smart method.     

两类求解y″＝f（x，y）具有极小相位延迟的显式方法

本文提出了两类数值积分二阶周期性初值问题ｙ″＝ｆ（ｘ，ｙ），ｙ（ｘ０）＝ｙ０，ｙ＇（ｘ０）＝ｙ＇０具有极小相位延迟的显式两步法。这些方法推广和改进了文献［１］－［７］中的某些方法。数值试验表明本文中的某些方法优于［１］－［７］中的某些方法。     

A two-step explicit P-stable method of high phase-lag order for linear periodic IVPs

In this paper, we present a nonlinear two-step explicit P-stable method of fourth algebraic order and 12th phase-lag order for solving one-dimensional second-order linear periodic initial value problems (IVPs) of ordinary differential equations. Based on a special vector arithmetic with respect to an analytic function, the method can be extended to be vector-applicable for multi-dimensional problems directly. Some numerical results are reported to illustrate the efficiency of the method.     

A new methodology for the construction of numerical methods for the approximate solution of the Schr?dinger equation

In the present paper we introduce a new methodology for the development of numerical methods for the numerical solution of the one-dimensional Schr?dinger equation. The new methodology is based on the requirement of vanishing the phase-lag and its derivatives. The efficiency of the new methodology is proved via error analysis and numerical applications. T. E. Simos is Highly Cited Researcher, Active Member of the European Academy of Sciences and Arts. Corresponding Member of the European Academy of Sciences and European Academy of Arts, Sciences and Humanities.     

Symmetric multistep Obrechkoff methods with zero phase-lag for periodic initial value problems of second order differential equations

In this paper, symmetric multistep Obrechkoff methods of orders 8 and 12, involving a parameter p to solve a special class of second order initial value problems in which the first order derivative does not appear explicitly, are discussed. It is shown that the methods have zero phase-lag when p is chosen as 2π times the frequency of the given initial value problem.