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.

An improved trigonometrically fitted P-stable Obrechkoff method for periodic initial-value problems

2005 ◽  
Vol 461 (2058) ◽  
pp. 1639-1658 ◽  
Author(s):  
Zhongcheng Wang ◽  
Deying Zhao ◽  
Yongming Dai ◽  
Dongmei Wu
Keyword(s):  
Initial Value Problems ◽  
New Method ◽  
Order Derivative ◽  
Phase Lag ◽  
Initial Value ◽  
Numerical Illustration ◽  
First Order ◽  
Derivative Formula ◽  
Periodic Initial Value Problems

In this paper we present an improved P-stable trigonometrically fitted Obrechkoff method with phase-lag (frequency distortion) infinity. Compared with the previous P-stable trigonometrically fitted Obrechkoff method developed by Simos, our new method is simpler in structure and more stable in computation. We have also improved the accuracy of the first-order derivative formula. From the numerical illustration presented, we can show that the new method is much more accurate than the previous methods.

A P-STABLE EIGHTEENTH-ORDER SIX-STEP METHOD FOR PERIODIC INITIAL VALUE PROBLEMS

2007 ◽  
Vol 18 (03) ◽  
pp. 419-431 ◽  
Author(s):  
CHUNFENG WANG ◽  
ZHONGCHENG WANG
Keyword(s):  
Initial Value Problems ◽  
Numerical Results ◽  
New Method ◽  
Initial Value ◽  
Step Method ◽  
Periodic Initial Value Problems

In this paper we present a new kind of P-stable eighteenth-order six-step method for periodic initial-value problems. We add the fourth derivatives to our previous P-stable six-step method to increase the accuracy. We apply two classes of well-known problems to our new method and compare it with the previous methods. The numerical results show that the new method is much more stable, accurate and efficient than the previous methods.

A P-stable complete in phase Obrechkoff trigonometric fitted method for periodic initial-value problems

1993 ◽  
Vol 441 (1912) ◽  
pp. 283-289 ◽  
Keyword(s):  
Initial Value Problems ◽  
Numerical Results ◽  
New Method ◽  
Phase Lag ◽  
Initial Value ◽  
Periodic Initial Value Problems

In this paper we derive a P-stable trigonometric fitted Obrechkoff method with phase-lag (frequency distortion) infinity. It is easy to see, from numerical results presented, that the new method is much more accurate than previous methods.

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.

A PHASE-FITTED AND AMPLIFICATION-FITTED EXPLICIT TWO-STEP HYBRID METHOD FOR SECOND-ORDER PERIODIC INITIAL VALUE PROBLEMS

2006 ◽  
Vol 17 (05) ◽  
pp. 663-675 ◽  
Author(s):  
HANS VAN DE VYVER
Keyword(s):  
Hybrid Method ◽  
Initial Value Problems ◽  
Numerical Results ◽  
New Method ◽  
Test Problems ◽  
Phase Lag ◽  
Initial Value ◽  
Periodic Initial Value Problems ◽  
Free Parameters ◽  
Fifth Order

In this paper a phase-fitted and amplification-fitted explicit two-step hybrid method is introduced. The construction is based on a modification of a fifth-order dissipative method recently developed by Franco.19 Two free parameters are added in order to nullify the phase-lag and the amplification. Numerical results obtained for well-known test problems show the efficiency of the new method when it is compared with other existing codes.

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.

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