Construction of exponentially fitted symplectic Runge-Kutta-Nyström methods from partitioned Runge-Kutta methods

DOI10.1007/s00009-015-0587-2zbMath1347.65123OpenAlexW964683004MaRDI QIDQ305895

F. Blanchet-Sadri, M. Dambrine

Publication date: 31 August 2016

Published in: Mediterranean Journal of Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00009-015-0587-2

zbMATH Keywords

Hamiltonian systemsRunge-Kutta-Nyström methodssymplectic methodspartitioned Runge-Kutta methodsnumerical resulttrigonometrical fitting

Mathematics Subject Classification ID

Nonlinear ordinary differential equations and systems (34A34) Numerical methods for initial value problems involving ordinary differential equations (65L05) Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations (65L06) Numerical methods for Hamiltonian systems including symplectic integrators (65P10) Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems (37M15)

Related Items (91)

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 ⋮ New hybrid two-step method with optimized phase and stability characteristics ⋮ New Runge-Kutta type symmetric two-step method with optimized characteristics ⋮ A new fourteenth algebraic order finite difference method for the approximate solution of the Schrödinger equation ⋮ An economical eighth-order method for the approximation of the solution of the Schrödinger equation ⋮ 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 new implicit high-order six-step singularly P-stable method for the numerical solution of Schrödinger equation ⋮ 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 ⋮ On ninth order, explicit Numerov-type methods with constant coefficients ⋮ New five-stages finite difference pair with optimized phase properties ⋮ 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 ⋮ A five-stages symmetric method with improved phase properties ⋮ A new eighth order exponentially fitted explicit Numerov-type method for solving oscillatory problems ⋮ New five-stages two-step method with improved characteristics ⋮ A new high algebraic order four stages symmetric 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 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 ⋮ On the structure of discrete spectrum of a non-selfadjoint system of differential equations with integral boundary condition ⋮ 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 ⋮ An efficient and computational effective method for second order problems ⋮ Stability and convergence of an effective finite element method for multiterm fractional partial differential equations ⋮ New multiple stages two-step complete in phase algorithm with improved characteristics for second order initial/boundary value problems ⋮ New multiple stages multistep method with best possible phase properties for second order initial/boundary value problems ⋮ New four stages multistep in phase algorithm with best possible properties for second order problems ⋮ New multistage two-step complete in phase scheme with improved properties for quantum chemistry problems ⋮ A new multistage multistep full in phase algorithm with optimized characteristics for problems in chemistry ⋮ A new four-stages two-step phase fitted scheme for problems in quantum chemistry ⋮ 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 ⋮ Evolutionary derivation of sixth-order P-stable SDIRKN methods for the solution of PDEs with the method of lines ⋮ An efficient and economical high order method for the numerical approximation of the Schrödinger equation ⋮ A generator of families of two-step numerical methods with free parameters and minimal phase-lag ⋮ High-order exponentially fitted and trigonometrically fitted explicit two-derivative Runge-Kutta-type methods for solving third-order oscillatory problems ⋮ The new class of multistep multiderivative hybrid methods for the numerical solution of chemical stiff systems of first order IVPs ⋮ A multistep method with optimal properties for second order differential equations ⋮ A four stages numerical pair with optimal phase and stability properties ⋮ A finite difference pair with improved phase and stability properties ⋮ 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 hybrid finite difference pair with maximum phase and stability properties ⋮ New finite difference pair with optimized phase and stability properties ⋮ New four-stages symmetric six-step method with improved phase properties for second order problems with periodical and/or oscillating solutions ⋮ New Runge-Kutta type symmetric two step finite difference pair with improved properties for second order initial and/or boundary value problems ⋮ A new multistep method with optimized characteristics for initial and/or boundary value problems ⋮ New multiple stages scheme with improved properties for second order problems ⋮ 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 ⋮ A new family of three-stage two-step P-stable multiderivative methods with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrödinger equation and IVPs with oscillating solutions ⋮ 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 ⋮ Trigonometrically fitted high-order predictor-corrector method with phase-lag of order infinity for the numerical solution of radial Schrödinger equation ⋮ 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 ⋮ New three-stages symmetric two step method with improved properties for second order initial/boundary value problems ⋮ New 8-step symmetric embedded predictor-corrector (EPCM) method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation ⋮ New hybrid symmetric two step scheme with optimized characteristics for second order problems ⋮ 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 ⋮ SYMPLECTIC RUNGE-KUTTA METHODS OF HIGH ORDER BASED ON W-TRANSFORMATION ⋮ 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 low computational cost eight algebraic order hybrid method with vanished phase-lag and its first, second, third and fourth derivatives for the approximate solution of the Schrödinger equation ⋮ Exponentially fitted TDRK pairs for the Schrödinger equation ⋮ A new multistep finite difference pair for the Schrödinger equation and related problems ⋮ A new two-step finite difference pair with optimal properties ⋮ An efficient numerical method for the 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 ⋮ 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 ⋮ A three-stages multistep teeming in phase algorithm for computational problems in chemistry ⋮ A four-stages multistep fraught in phase method for quantum chemistry problems ⋮ Explicit, eighth-order, four-step methods for solving \(y^{\prime\prime}=f(x, y)\) ⋮ Explicit Runge-Kutta methods for starting integration of Lane-Emden problem ⋮ A new explicit four-step method with vanished phase-lag and its first and second derivatives ⋮ Hybrid Numerov-type methods with coefficients trained to perform better on classical orbits ⋮ Trigonometric-fitted explicit Numerov-type method with vanishing phase-lag and its first and second derivatives ⋮ Neural network solution of single-delay differential equations ⋮ Eighth order, phase-fitted, six-step methods for solving \(y^{\prime \prime}=f(x,y)\) ⋮ Algorithm for the development of families of numerical methods based on phase-lag Taylor series ⋮ A multistage two-step fraught in phase scheme for problems in mathematical chemistry ⋮ A Runge-Kutta type crowded in phase algorithm for quantum chemistry problems ⋮ A perfect in phase FD algorithm for problems in quantum chemistry ⋮ A multiple stage absolute in phase scheme for chemistry problems ⋮ Bounds for variable degree rational \(L_\infty\) approximations to the matrix exponential ⋮ Unnamed Item ⋮ A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation ⋮ 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

Cites Work

• Unnamed Item
• Parameter estimation in exponentially fitted hybrid methods for second order differential problems
• Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems
• Symplectic conditions for exponential fitting Runge-Kutta-Nyström methods
• An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions
• Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials
• Frequency evaluation in exponential fitting multistep algorithms for ODEs
• Exponentially-fitted Runge-Kutta-Nyström method for the numerical solution of initial-value problems with oscillating solutions
• Frequency evaluation for exponentially fitted Runge-Kutta methods
• Symplectic Partitioned Runge-Kutta Methods for the Numerical Integration of Periodic and Oscillatory Problems
• On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods
• A Symplectic Trigonometrically Fitted Modified Partitioned Runge-Kutta Method for the Numerical Integration of Orbital Problems

This page was built for publication: Construction of exponentially fitted symplectic Runge-Kutta-Nyström methods from partitioned Runge-Kutta methods