New two stages high order symmetric six-step method with vanished phase-lag and its first, second and third derivatives for the numerical solution of the Schrödinger equation

DOI10.1007/s10910-016-0691-yzbMath1384.65047OpenAlexW2528609040MaRDI QIDQ1704722

Ibraheem Alolyan, Theodore E. Simos

Publication date: 13 March 2018

Published in: Journal of Mathematical Chemistry (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10910-016-0691-y

zbMATH Keywords

stabilityfinite difference methodSchrödinger equationphase-lagmultistep methodsnumerical resultderivatives of the phase-laginterval of periodicitymultistage methodsphase-fitted

Mathematics Subject Classification ID

Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) (34L40) Stability and convergence of numerical methods for ordinary differential equations (65L20) Numerical methods for initial value problems involving ordinary differential equations (65L05) Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations (65L06) Finite difference and finite volume methods for ordinary differential equations (65L12)

Related Items (11)

New three-stages symmetric six-step finite difference method with vanished phase-lag and its derivatives up to sixth derivative for second order initial and/or boundary value problems with periodical and/or oscillating solutions ⋮ A new six-step algorithm with improved properties for the numerical solution of second order initial and/or boundary value problems ⋮ A new three-stages six-step finite difference pair with optimal phase properties for second order initial and/or boundary value problems with periodical and/or oscillating solutions ⋮ Three stages symmetric six-step method with eliminated phase-lag and its derivatives for the solution of the Schrödinger equation ⋮ An efficient six-step method for the solution of the Schrödinger equation ⋮ New multiple stages multistep method with best possible phase properties for second order initial/boundary value problems ⋮ A multistep method with optimal properties for second order differential equations ⋮ New four-stages symmetric six-step method with improved phase properties for second order problems with periodical and/or oscillating solutions ⋮ A new multistep finite difference pair for the Schrödinger equation and related problems ⋮ High order computationally economical six-step method with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation ⋮ Algorithm for the development of families of numerical methods based on phase-lag Taylor series

Cites Work

• Families of Runge-Kutta-Nystrom Formulae
• Symmetric Multistip Methods for Periodic Initial Value Problems
• A NEW METHODOLOGY FOR THE CONSTRUCTION OF OPTIMIZED RUNGE–KUTTA–NYSTRÖM METHODS
• TWO NEW PHASE-FITTED SYMPLECTIC PARTITIONED RUNGE–KUTTA METHODS
• On finite difference methods for the solution of the Schrödinger equation
• A Fifth-order Symplectic Trigonometrically Fitted Partitioned Runge-Kutta Method
• CLOSED NEWTON–COTES TRIGONOMETRICALLY-FITTED FORMULAE FOR LONG-TIME INTEGRATION
• A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation
• A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrödinger equation. I: Development of the basic method
• A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrödinger equation. II: Development of the generator, optimization of the generator and numerical results
• A modified phase-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• A new approach on the construction of trigonometrically fitted two step hybrid methods
• An implicit symmetric linear six-step methods with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation and related problems
• A new two stage symmetric two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation
• 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
• A family of embedded explicit six-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation: development and theoretical analysis
• A new four stages symmetric two-step method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation
• Construction of exponentially fitted symplectic Runge-Kutta-Nyström methods from partitioned Runge-Kutta methods
• An optimized two-step hybrid block method for solving general second order initial-value problems
• A family of two stages tenth algebraic order symmetric six-step methods with vanished phase-lag and its first derivatives for the numerical solution of the radial Schrödinger equation and related problems
• 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
• New open modified Newton Cotes type formulae as multilayer symplectic integrators
• A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems
• A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems
• New stable closed Newton-Cotes trigonometrically fitted formulae for long-time integration
• Optimizing a hybrid two-step method for the numerical solution of the Schrödinger equation and related problems with respect to phase-lag
• An explicit four-step method with vanished phase-lag and its first and second derivatives
• A Runge-Kutta type four-step method with vanished phase-lag and its first and second derivatives for each level for the numerical integration of the Schrödinger equation
• A new explicit hybrid four-step method with vanished phase-lag and its derivatives
• An explicit linear six-step method with vanished phase-lag and its first derivative
• A family of explicit linear six-step methods with vanished phase-lag and its first derivative
• A hybrid type four-step method with vanished phase-lag and its first, second and third derivatives for each level for the numerical integration of the Schrödinger equation
• An eight-step semi-embedded predictor-corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown
• A high algebraic order predictor-corrector explicit method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation and related problems
• Efficient low computational cost hybrid explicit four-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the Schrödinger equation
• 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
• Runge-Kutta type methods with special properties for the numerical integration of ordinary differential equations
• A fourth order modified trigonometrically fitted symplectic Runge-Kutta-Nyström method
• Symplectic partitioned Runge-Kutta methods with minimal phase-lag
• A nonlinear explicit two-step fourth algebraic order method of order infinity for linear periodic initial value problems
• A family of four-step trigonometrically-fitted methods and its application to the Schrödinger equation
• Closed Newton-Cotes trigonometrically-fitted formulae of high order for the numerical integration of the Schrödinger equation
• Computation of the eigenvalues of the Schrödinger equation by symplectic and trigonometrically fitted symplectic partitioned Runge-Kutta methods
• New modified Runge-Kutta-Nyström methods for the numerical integration of the Schrödinger equation
• Construction of an optimized explicit Runge-Kutta-Nyström method for the numerical solution of oscillatory initial value problems
• A two-step method with vanished phase-lag and its first two derivatives for the numerical solution of the Schrödinger equation
• On modified Runge-Kutta trees and methods
• P-stable linear symmetric multistep methods for periodic initial-value problems
• Families of third and fourth algebraic order trigonometrically fitted symplectic methods for the numerical integration of Hamiltonian systems
• High-order closed Newton-Cotes trigonometrically-fitted formulae for long-time integration of orbital problems
• High order closed Newton-Cotes exponentially and trigonometrically fitted formulae as multilayer symplectic integrators and their application to the radial Schrödinger equation
• Closed Newton-Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems
• A family of two-stage two-step methods for the numerical integration of the Schrödinger equation and related IVPs with oscillating solution
• Two optimized symmetric eight-step implicit methods for initial-value problems with oscillating solutions
• An optimized explicit Runge-Kutta method with increased phase-lag order for the numerical solution of the Schrödinger equation and related problems
• A new two-step hybrid method for the numerical solution of the Schrödinger equation
• Trigonometrically fitted and exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation
• The numerical solution of the radial Schrödinger equation via a trigonometrically fitted family of seventh algebraic order predictor-corrector methods
• A four-step exponentially fitted method for the numerical solution of the Schrödinger equation
• A family of exponentially-fitted Runge-Kutta methods with exponential order up to three for the numerical solution of the Schrödinger equation
• A new family of two stage symmetric two-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation
• Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation
• A family of trigonometrically fitted partitioned Runge-Kutta symplectic methods
• High order closed Newton-Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation
• A new Numerov-type method for the numerical solution of the Schrödinger equation
• A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II: Explicit method
• A family of embedded Runge-Kutta formulae
• A finite-difference method for the numerical solution of the Schrödinger equation
• Newton--Cotes formulae for long-time integration.
• Symplectic integrators for the numerical solution of the Schrödinger equation
• A generator of hybrid symmetric four-step methods for the numerical solution of the Schrödinger equation
• Trigonometrically fitted predictor--corrector methods for IVPs with oscillating solutions
• A family of trigonometrically-fitted symmetric methods for the efficient solution of the Schrödinger equation and related problems
• Symplectic methods for the numerical solution of the radial Schrödinger equation
• Special issue: Selected papers from the conference on computational and mathematical methods for science and engineering (CMMSE-2002), Alicante, Spain, 20--25 September 2002
• Zero dissipative, explicit Numerov-type methods for second order IVPs with oscillating solutions
• Symplectic methods of fifth order for the numerical solution of the radial Schrödinger equation
• Exponentially-fitted Runge-Kutta-Nyström method for the numerical solution of initial-value problems with oscillating solutions
• Symmetric eighth algebraic order methods with minimal phase-lag for the numerical solution of the Schrödinger equation
• Construction of trigonometrically and exponentially fitted Runge--Kutta--Nyström methods for the numerical solution of the Schrödinger equation and related problems -- a method of 8th algebraic order
• Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution.
• An optimized Runge-Kutta method for the solution of orbital problems
• A fourth algebraic order trigonometrically fitted predictor-corrector scheme for IVPs with oscillating solutions
• Multiderivative methods of eighth algebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation
• Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics
• An explicit sixth-order method with phase-lag of order eight for \(y=f(t,y)\)
• Comparison of some four-step methods for the numerical solution of the Schrödinger equation
• Optimized Runge-Kutta pairs for problems with oscillating solutions
• New P-stable eighth algebraic order exponentially-fitted methods for the numerical integration of the Schrödinger equation
• Family of twelve steps exponential fitting symmetric multistep methods for the numerical solution of the Schrödinger equation
• Exponentially-fitted multiderivative methods for the numerical solution of the Schrödinger equation
• New high order multiderivative explicit four-step methods with vanished phase-lag and its derivatives for the approximate solution of the Schrödinger equation. I: construction and theoretical analysis
• A symmetric eight-step predictor-corrector method for the numerical solution of the radial Schrödinger equation and related IVPs with oscillating solutions
• A new explicit four-step method with vanished phase-lag and its first and second derivatives
• Optimization as a function of the phase-lag order of nonlinear explicit two-step \(P\)-stable method for linear periodic IVPs
• A Runge-Kutta type implicit high algebraic order two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation
• A predictor-corrector explicit four-step method with vanished phase-lag and its first, second and third derivatives for the numerical integration of the Schrödinger equation
• An explicit Numerov-type method for second-order differential equations with oscillating solutions
• A trigonometrically fitted explicit hybrid method for the numerical integration of orbital problems
• A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation
• A new four-step hybrid type method with vanished phase-lag and its first derivatives for each level for the approximate integration of the Schrödinger equation
• New second-order exponentially and trigonometrically fitted symplectic integrators for the numerical solution of the time-independent Schrödinger equation
• Phase-fitted and amplification-fitted two-step hybrid methods for \(y^{\prime\prime }=f(x,y)\)
• Exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation
• Trigonometrically fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation
• Sixth algebraic order trigonometrically fitted predictor-corrector methods for the numerical solution of the radial Schrödinger equation
• A family of multiderivative methods for the numerical solution of the Schrödinger equation
• High-order predictor--corrector of exponential fitting for the \(N\)-body problems
• Special issue: The international conference on computational methods in sciences and engineering 2004 (ICCMSE-2004), Vouliagmeni, Greece, November 19--23, 2004. Selected papers based on the presentation at the conference.
• A fourth algebraic order exponentially-fitted Runge-Kutta method for the numerical solution of the Schrodinger equation
• A NEW SYMMETRIC EIGHT-STEP PREDICTOR-CORRECTOR METHOD FOR THE NUMERICAL SOLUTION OF THE RADIAL SCHRÖDINGER EQUATION AND RELATED ORBITAL PROBLEMS

This page was built for publication: New two stages high order symmetric six-step method with vanished phase-lag and its first, second and third derivatives for the numerical solution of the Schrödinger equation