Symplectic integrators for the numerical solution of the Schrödinger equation
TLDR
In this paper, the Schrodinger equation is transformed into a Hamiltonian canonical equation and the concept of asymptotic symplecticness is introduced and methods of order up to 3 are developed.About:
This article is published in Journal of Computational and Applied Mathematics.The article was published on 2003-09-01 and is currently open access. It has received 151 citations till now. The article focuses on the topics: Symplectic integrator & Variational integrator.read more
Citations
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Journal ArticleDOI
A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems
Zacharias Anastassi,T. E. Simos +1 more
TL;DR: An explicit symmetric linear phase-fitted four-step method with a free coefficient as parameter used for the optimization of the method in order to solve efficiently the Schrodinger equation and related oscillatory problems is developed.
Journal ArticleDOI
Optimizing a Hybrid Two-Step Method for the Numerical Solution of the Schrödinger Equation and Related Problems with Respect to Phase-Lag
TL;DR: This work studies how the vanishing of the phase-lag and its derivatives optimizes the efficiency of the hybrid two-step method for the numerical solution of the radial Schrodinger equation and related problems with periodic or oscillating solutions.
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A Modified Runge-Kutta-Nyström Method by using Phase Lag Properties for the Numerical Solution of Orbital Problems
D. F. Papadopoulos,T. E. Simos +1 more
TL;DR: In this paper, a modified Runge-Kutta-Nystrom method of third algebraic order is developed, which has phase-lag and amplification error of order infinity.
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A family of trigonometrically fitted partitioned Runge–Kutta symplectic methods
TL;DR: A family of trigonometrically fitted partitioned Runge–Kutta symplectic methods of fourth order with six stages considering the solution of the one-dimensional time independent Schrodinger equation is presented.
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On the Explicit Four-Step Methods with Vanished Phase-Lag and its First Derivative
T. E. Simos,Saudi Arabia +1 more
TL;DR: In this paper, the authors investigated a family of explicit four-step methodel methods for the case of vanishing of phase-lag and its first derivative, which are efficient for the numerical solution of the Schr ¨ odinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions.
References
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Book
Mathematical Methods of Classical Mechanics
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Journal ArticleDOI
Construction of higher order symplectic integrators
TL;DR: For Hamiltonian systems of the form H = T(p)+V(q) a method was shown to construct explicit and time reversible symplectic integrators of higher order as discussed by the authors.
Book
Numerical Hamiltonian Problems
TL;DR: Examples of Hamiltonian Systems, symplectic integration, and Numerical Methods: Checking preservation of area: Jacobians, and Necessity of the symplecticness conditions.
Journal ArticleDOI
Practical points concerning the solution of the Schrödinger equation
John M. Blatt,John M. Blatt +1 more
TL;DR: In this article, the numerical solution of the one-dimensional Schrodinger equation in a potential of the type occurring in molecular spectroscopy was considered, with both an inner and an outer classical turning point.
Journal ArticleDOI
Two-step methods for the numerical solution of the Schrödinger equation
TL;DR: A new two-step exponentially-fitted formula is derived and applied to the Schrödinger equation and is found to be significantly more accurate than the standard methods, for large values of the energy.
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