Book ChapterDOI
New Families of Symplectic Runge-Kutta-Nyström Integration Methods
Sergio Blanes,Fernando Casas,José Luis Regidor Ros +2 more
- pp 102-109
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TLDR
New 6-th and 8-th order explicit symplectic Runge-Kutta-Nystrom methods for Hamiltonian systems which are more efficient than other previously known algorithms are presented.Abstract:
We present new 6-th and 8-th order explicit symplectic Runge-Kutta-Nystrom methods for Hamiltonian systems which are more efficient than other previously known algorithms. The methods use the processing technique and non-trivial flows associated with different elements of the Lie algebra involved in the problem. Both the processor and the kernel are compositions of explicitly computable maps.read more
Citations
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Splitting and composition methods in the numerical integration of differential equations
TL;DR: A comprehensive survey of splitting and composition methods for numerical integration of ODEs can be found in this paper, where the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable.
Journal ArticleDOI
A new phase-fitted eight-step symmetric embedded predictor–corrector method (EPCM) for orbital problems and related IVPs with oscillating solutions
TL;DR: The new phase-fitted embedded predictor–corrector method (EPCM) presented here is based on the multistep symmetric method of Quinlan–Tremaine (1990), with eight steps and eighth algebraic order and constructed to solve numerically the two-dimensional Kepler problem.
Journal ArticleDOI
Explicit adaptive symplectic (easy) integrators: a scaling invariant generalisation of the levi-civita and ks regularisations
Sergio Blanes,Chris Budd +1 more
TL;DR: In this article, a generalisation of the Levi-Civita and Kustaanheimo-Stiefel regularisation is presented, which allows the use of more general time rescalings.
Journal ArticleDOI
The construction of arbitrary order ERKN methods based on group theory for solving oscillatory Hamiltonian systems with applications
Lijie Mei,Xinyuan Wu +1 more
TL;DR: High-order structure-preserving ERKN methods are obtained in an effective way for solving oscillatory Hamiltonian systems using extended Runge-Kutta-Nystrom methods.
Journal ArticleDOI
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
TL;DR: In this article, a new embedded predictor-corrector phase-fitted method with vanished phase-lag is developed for the first time in literature, which can be used to solve numerically IVPs with oscillatory solutions, orbital problems and the Schrodinger equation.
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
Lie groups, Lie algebras, and their representations
TL;DR: In this article, differentiable and analytic manifolds and Lie Groups and Lie Algebras have been studied in the context of structure theory and representation theory, and complex semisimple Lie Algebraic structures have been proposed.
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.