A multistep method with optimal properties for second order differential equations
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Publication:1703391
DOI10.1007/s10910-017-0774-4zbMath1384.65049OpenAlexW2735268187MaRDI QIDQ1703391
Maxim A. Medvedev, Theodore E. Simos
Publication date: 2 March 2018
Published in: Journal of Mathematical Chemistry (Search for Journal in Brave)
Full work available at URL: https://doi.org/10.1007/s10910-017-0774-4
algorithmstabilitynumerical examplefinite difference methodSchrödinger equationphase-lagcomputational efficiencymultistep methodslocal truncation errorderivatives of the phase-laginterval of periodicitymultistage methodsphase-fitted
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) Error bounds for numerical methods for ordinary differential equations (65L70) Finite difference and finite volume methods for ordinary differential equations (65L12)
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Cites Work
- 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
- Families of Runge-Kutta-Nystrom Formulae
- Evolutionary generation of high-order, explicit, two-step methods for second-order linear IVPs
- Symmetric Multistip Methods for Periodic Initial Value Problems
- Modified two‐step hybrid methods for the numerical integration of oscillatory 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 new high algebraic order efficient finite difference method 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
- 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 high-order two-step phase-fitted method for the numerical solution of the Schrödinger equation
- The use of phase lag and amplification error derivatives for the construction of a modified Runge-Kutta-Nyström method
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Phase-fitted Runge-Kutta pairs of orders 8(7)
- 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
- 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
- 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
