On the Explicit Four-Step Methods with Vanished Phase-Lag and its First Derivative

Abstract

Abstract:In the present paper, we will investigate a family of explicit four-step methods first introduced by Anastassi and Simos [1] for the case of vanishing of phase-lag and its first derivative. These methods are efficient for the numerical solution of the Schrödinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions. As we mentioned before the main scope of this paper is the study of the elimination of the phase-lag and its first derivative of the family of the methods to be investigated. A comparative error and stability analysis will be presented for the studied family of four-step explicit methods. The new obtained methods will finally be tested on the resonance problem of the Schrödinger equation in order to examine their efficiency.

A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation

“…The Method developed in [45], which is indicated as Method RKTPLDDEA is more efficient than the method developed in [52] (Case 1), which is indicated as Method NMC1. 6.…”

“…9 Accuracy (digits) for several values of C PU time (in seconds) for the eigenvalue E 2 = 341.495874. The nonexistence of a value of accuracy (digits) indicates that for this value of CPU, accuracy (digits) is less than 0 -The Phase-Fitted Method (Case 2) developed in [48], which is indicated as Method NMPF2 -The Method developed in [52] (Case 2), which is indicated as Method NMC2 -The Method developed in [52] (Case 1), which is indicated as Method NMC1 -The Method developed in [45], which is indicated as Method RKTPLDDEA -The Method developed in [58], which is indicated as Method HYBPLDDDEA -The Hybrid Low Computational Computational Cost Four-Step Method developed in [46], which is indicated as Method HYMETH8 -The New Obtained Three Stages Explicit Symmetric Four-Step Method which is developed in Sect. 5, which is indicated as Method MuSMeth10…”

“…The twelfth algebraic order multistep method developed by Quinlan and Tremaine [22], which is indicated as Method QT12 is more efficient than the tenth order multistep method developed by Quinlan and Tremaine [22], which is indicated as Method QT10 4. The Method developed in [52] (Case 1), which is indicated as Method NMC1…”

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

“…-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…”

A New Optimized Symmetric Embedded Predictor- Corrector Method (EPCM) for Initial-Value Problems with Oscillatory Solutions

“…They have constructed high order symmetric multistep methods based on the work of Lambert and Watson (see [2]). Many numerical methods have been developed for the numerical solution of the initial value problem (1) (see [17] - [19] and [21]- [25]) …”