A FAMILY OF HYBRID EIGHTH ORDER METHODS WITH MINIMAL PHASE-LAG FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION AND RELATED PROBLEMS

2000 ◽  
Vol 11 (02) ◽  
pp. 415-437 ◽  
Author(s):  
G. AVDELAS ◽  
A. KONGUETSOF ◽  
T. E. SIMOS
Keyword(s):  
Numerical Solution ◽  
Initial Value Problems ◽  
Hybrid Methods ◽  
Phase Lag ◽  
Initial Value ◽  
Eighth Order ◽  
New Methods ◽  
Algebraic Order ◽  
Periodic Initial Value Problems ◽  
Minimal Phase

In this paper a family of hybrid methods with minimal phase-lag are developed for the numerical solution of periodic initial-value problems. The methods are of eighth algebraic order and have large intervals of periodicity. The efficiency of the new methods is presented by their application to the wave equation and to coupled differential equations of the Schrödinger type.

EXPLICIT EIGHTH ORDER NUMEROV-TYPE METHODS WITH REDUCED NUMBER OF STAGES FOR OSCILLATORY IVPs

2008 ◽  
Vol 19 (06) ◽  
pp. 957-970 ◽  
Author(s):  
I. Th. FAMELIS
Keyword(s):  
Initial Value Problems ◽  
Computational Cost ◽  
Phase Lag ◽  
Initial Value ◽  
Eighth Order ◽  
Oscillatory Solutions ◽  
Numerical Tests ◽  
New Methods ◽  
Oscillating Solutions ◽  
Algebraic Order

Using a new methodology for deriving hybrid Numerov-type schemes, we present new explicit methods for the solution of second order initial value problems with oscillating solutions. The new methods attain algebraic order eight at a cost of eight function evaluations per step which is the most economical in computational cost that can be found in the literature. The methods have high amplification and phase-lag order characteristics in order to suit to the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our effort.

AN EXPLICIT HIGH ORDER PREDICTOR-CORRECTOR METHOD FOR PERIODIC INITIAL VALUE PROBLEMS

1995 ◽  
Vol 05 (02) ◽  
pp. 159-166 ◽  
Author(s):  
T.E. SIMOS
Keyword(s):  
Initial Value Problems ◽  
Interval Of Periodicity ◽  
Phase Lag ◽  
Initial Value ◽  
Large Interval ◽  
Type Method ◽  
Algebraic Order ◽  
Periodic Initial Value Problems ◽  
Orbit Equation ◽  
Predictor Corrector

An explicit Runge-Kutta type method is developed here. This method has an algebraic order six, a large interval of periodicity and a phase-lag of order eight. It is much more efficient than other well known methods when applying to an orbit equation.

A HIGHLY ACCURATE AND EFFICIENT TRIGONOMETRICALLY-FITTED P-STABLE THREE-STEP METHOD FOR PERIODIC INITIAL-VALUE PROBLEMS

2006 ◽  
Vol 17 (04) ◽  
pp. 545-560
Author(s):  
ZHONGCHENG WANG ◽  
YONGMING DAI ◽  
DONGMEI WU
Keyword(s):  
Initial Value Problems ◽  
The Other ◽  
New Method ◽  
Phase Lag ◽  
Initial Value ◽  
Step Method ◽  
Eighth Order ◽  
Numerical Illustration ◽  
Even Order ◽  
Periodic Initial Value Problems

In this paper we present a new three-step method, which is a trigonometrically-fitted P-stable Obrechkoff method with phase-lag (frequency distortion) infinity. In this new method, we make use of higher-even-order derivatives including the eighth-order to increase the accuracy. On the other hand, we adopt a special structure to reduce the computational complexity of high-derivatives. The numerical illustration demonstrate that the new method has advantage in accuracy, periodic stability and efficiency.

ON THE CONSTRUCTION OF EFFICIENT METHODS FOR SECOND ORDER IVPS WITH OSCILLATING SOLUTION

2001 ◽  
Vol 12 (10) ◽  
pp. 1453-1476 ◽  
Author(s):  
T. E. SIMOS ◽  
JESUS VIGO AGUIAR
Keyword(s):  
Initial Value Problems ◽  
Multistep Methods ◽  
Oscillating Solution ◽  
Second Order ◽  
Phase Lag ◽  
Initial Value ◽  
New Methods ◽  
Oscillating Solutions ◽  
Exponentially Fitted ◽  
Minimal Phase

In this paper we describe procedures for the construction of efficient methods for the numerical solution of second order initial value problems (IVPs) with oscillating solutions. Based on the described procedures we develop two simple and efficient multistep methods for the solution of the above problems. The first method is exponentially-fitted and trigonometrically-fitted and the second has a minimal phase-lag. Both methods are symmetric. Numerical results obtained for several well known problems show the efficiency of the new methods when they are compared with known methods in the literature.

Sign in / Sign up

Export Citation Format

Share Document