A four-step exponentially fitted method for the numerical solution of the Schrödinger equation

DOI10.1007/s10910-006-9170-1zbMath1125.81023OpenAlexW2091006293MaRDI QIDQ882577

Theodore E. Simos

Publication date: 24 May 2007

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

Full work available at URL: https://doi.org/10.1007/s10910-006-9170-1

zbMATH Keywords

stabilitySchrödinger equationlinear multistep methodsnumerical solutionexponential fittingtrigonometric fitting

Mathematics Subject Classification ID

Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) Computational methods for problems pertaining to quantum theory (81-08)

Related Items (61)

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 ⋮ 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 ⋮ 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 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 ⋮ A new methodology for the development of numerical methods for the numerical solution of the Schrödinger equation ⋮ A new methodology for the construction of numerical methods for the approximate solution of the Schrödinger equation ⋮ High order multistep methods with improved phase-lag characteristics for the integration of the Schrödinger equation ⋮ 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 ⋮ 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 ⋮ 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 ⋮ Three stages symmetric six-step method with eliminated phase-lag and its derivatives for the solution of the Schrödinger equation ⋮ A new optimized symmetric 8-step semi-embedded predictor-corrector method for the numerical solution of the radial Schrödinger equation and related orbital problems ⋮ An efficient six-step method for the solution of the Schrödinger equation ⋮ P-stability, trigonometric-fitting and the numerical solution of the radial Schrödinger equation ⋮ Mulitstep methods with vanished phase-lag and its first and second derivatives for the numerical integration of the Schrödinger equation ⋮ A new high order two-step method with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation ⋮ New multiple stages multistep method with best possible phase properties for second order initial/boundary value problems ⋮ High order four-step hybrid method with vanished phase-lag and its derivatives for the approximate 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 generator of families of two-step numerical methods with free parameters and minimal phase-lag ⋮ New open modified trigonometrically-fitted Newton-Cotes type multilayer symplectic integrators for the numerical solution of the Schrödinger equation ⋮ A multistep method with optimal properties for second order differential equations ⋮ 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 ⋮ A new hybrid two-step method with vanished phase-lag and its first and second derivatives for the numerical solution of the Schrödinger equation and related problems ⋮ A hybrid method with phase-lag and derivatives equal to zero for the numerical integration of the Schrödinger equation ⋮ New four-stages symmetric six-step method with improved phase properties for second order problems with periodical and/or oscillating solutions ⋮ Two-step high order hybrid explicit method 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 ⋮ 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 family of ten-step methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation ⋮ A two-step method with vanished phase-lag and its first two derivatives for the numerical solution of the Schrödinger equation ⋮ 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 ⋮ 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 ⋮ A new multistep finite difference pair for the Schrödinger equation and related problems ⋮ A new two stages tenth algebraic order symmetric six-step method with vanished phase-lag and its first and second derivatives for the solution of the radial Schrödinger equation and related problems ⋮ Two stages six-step method with eliminated phase-lag and its first, second, third and fourth derivatives for the approximation of the Schrödinger equation ⋮ High order computationally economical six-step method with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation ⋮ Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation ⋮ A new explicit four-step method with vanished phase-lag and its first and second derivatives ⋮ A family of eight-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation ⋮ 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 ⋮ Pseudospectral methods of solution of the Schrödinger equation ⋮ High order closed Newton-Cotes exponentially and trigonometrically fitted formulae as multilayer symplectic integrators and their application to the radial Schrödinger equation ⋮ A new four-step Runge-Kutta type method with vanished phase-lag and its first, second and third derivatives for the numerical solution of the Schrödinger equation ⋮ 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 family of Runge-Kutta methods with zero phase-lag and derivatives for the numerical solution of the Schrödinger equation and related problems ⋮ High order phase fitted multistep integrators for the Schrödinger equation with improved frequency tolerance ⋮ 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

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• A four-step phase-fitted method for the numerical integration of second order initial-value problems
• Unconditionally stable Noumerov-type methods for second order differential equations
• Numerov made explicit has better stability
• A variable step method for the numerical integration of the one- dimensional Schrödinger equation
• A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II: Explicit method
• Some embedded modified Runge-Kutta methods for the numerical solution of some specific Schrödinger equations
• Eighth order methods with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem
• 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
• A \(P\)-stable exponentially fitted method for the numerical integration of the Schrödinger equation
• Embedded eighth order methods for the numerical solution of the Schrödinger equation
• 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
• 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
• A family of \(P\)-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation
• Practical points concerning the solution of the Schrödinger equation
• The numerical solution of coupled differential equations arising from the Schrödinger equation
• Extended One-Step Methods: An Exponential Fitting Approach
• Partitioned Linear Multistep Method for Long Term Integration of the N-Body Problem
• High-Order Embedded Runge-Kutta-Nystrom Formulae
• Numerical Methods for y″ =f(x, y) via Rational Approximations for the Cosine
• Symmetric Multistip Methods for Periodic Initial Value Problems
• Exponentially-fitted algorithms: fixed or frequency dependent knot points?
• Numerical Solution of the two-dimensional time independent Schrödinger Equation by symplectic schemes
• Effective Numerical Approximation of Schrödinger type Equations through Multiderivative Exponentially-fitted Schemes
• Efficient Numerical Solution of Orbital Problems with the use of Symmetric Four-step Trigonometrically-fitted Methods
• On finite difference methods for the solution of the Schrödinger equation
• Trigonometrically and Exponentially fitted Symplectic Methods of third order for the Numerical Integration of the Schrödinger Equation
• SPECIAL OPTIMIZED RUNGE–KUTTA METHODS FOR IVPs WITH OSCILLATING SOLUTIONS
• A P-stable exponentially-fitted method for the numerical integration of the Schrödinger equation
• An Improved Eigenvalue Corrector Formula for Solving the Schrodinger Equation for Central Fields
• A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation
• A family of P-stable eighth algebraic order methods with exponential fitting facilities
• 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

This page was built for publication: A four-step exponentially fitted method for the numerical solution of the Schrödinger equation