Family of symmetric linear six-step methods with vanished phase-lag and its derivatives and their application to the radial Schrödinger equation and related problems

Alolyan, Ibraheem; Simos, T. E.

doi:10.1007/s10910-015-0572-9

Cited by 17 publications

(4 citation statements)

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“…3. The classical symmetric six-step method presented in [62], which is indicated as Method LS6SCL for small step sizes is more efficient than the exponentiallyfitted method of Raptis and Allison [100], which is indicated as Method RA.…”

Section: Discussionmentioning

confidence: 99%

“…The symmetric six-step method with vanished phase-lag and its first derivative developed in [62], which is indicated as Method LS6SPFD, is more efficient than the symmetric six-step phase-fitted method developed in [62], which is indicated as Method LS6SPF, and the twelfth order multi-step method developed by Quinlan and Tremaine [22], which is indicated as Method QT12 10. The symmetric six-step method with vanished phase-lag and its first and second derivatives developed in [62], which is indicated as Method LS6SPFDD, is more efficient than the symmetric six-step method with vanished phase-lag and its first derivative developed in [62], which is indicated as Method LS6SPFD. 11.…”

Section: Discussionmentioning

confidence: 99%

“…13. The embedded six-step method with vanished phase-lag and its first, second and third derivatives developed in [61], which is indicated as Method EM6SPFDDD is more efficient than the symmetric six-step method with vanished phase-lag and its first, second and third derivatives developed in [62], which is indicated as Method LS6SPFD3 14. The symmetric six-step method with vanished phase-lag and its first, second, third and fourth derivatives developed in [63], which is indicated as Method LS6SPFD4, is the more efficient than all the methods mentioned above.…”

Section: Discussionmentioning

confidence: 99%

“…-The twelfth order multi-step method developed by Quinlan and Tremaine [22], which is indicated as Method QT12. -The fourth algebraic order method of Chawla and Rao with minimal phase-lag [27], which is indicated as Method MCR4 -The exponentially-fitted method of Raptis and Allison [100], which is indicated as Method RA -The hybrid sixth algebraic order method developed by Chawla and Rao with minimal phase-lag [26], which is indicated as Method MCR6 [48], which is indicated as Method NMPF1 -The Phase-Fitted Method (Case 2) developed in [48], which is indicated as Method NMPF2 -The Method developed in [52] (Case 1), which is indicated as Method NMC1 -The classical symmetric six-step method presented here, which is indicated as Method LS6SCL -The symmetric six-step phase-fitted method presented in [62], which is indicated as Method LS6SPF -The symmetric six-step method with vanished phase-lag and its first derivative presented in [62], which is indicated as Method LS6SPFD -The symmetric six-step method with vanished phase-lag and its first and second derivatives presented in [62], which is indicated as Method LS6SPFDD -The symmetric six-step method with vanished phase-lag and its first, second and third derivatives presented in [62], which is indicated as Method LS6SPFD3 -The symmetric six-step method with vanished phase-lag and its first, second, third and fourth derivatives presented in [63], which is indicated as Method LS6SPFD4 -The classical embedded symmetric six-step method presented in [61], which is indicated as Method EM6SCL -The embedded symmetric six-step method with vanished phase-lag and its first and second derivatives presented in [61], which is indicated as Method EM6SPFDD…”

Section: The Radial Schrödinger Equation and The Resonance Problemmentioning

confidence: 99%

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A new eight algebraic order embedded explicit six-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation

Alolyan

Simos

2016

J Math Chem

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A new embedded symmetric six-step method with vanished phase-lag and its first, second, third and fourth derivatives is obtained for the first time in the literature in this paper. More precisely, for this new method we will investigate: -The development of the methods, i.e., the definition of the coefficients of the new method in order the phase-lag and its first, second, third and fourth derivatives of the phase-lag to be vanished, -The computation of the formulae of the Local Truncation Error (LTE) of the methods of the new family of methods, -The computation of the asymptotic forms of the LTE by applying of the new developed method on a scalar test problem (which is the time independent radial Schrödinger equation), -The computation of the asymptotic forms of the LTE of other methods of the same family and the comparative local truncation error analysis of all the methods of the family, -The stability (interval of periodicity) analysis of the new obtained method. To study the stability, we will use a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis.123 J Math Chem -The behavior of the new produced method by applying them to the numerical solution of the resonance problem of the radial Schrödinger equation. We will prove its efficiency by comparing it with (1) well known methods of the literature and (2) very recently obtained methods.

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Section: Discussionmentioning

confidence: 99%