# New families of symplectic splitting methods for numerical integration in dynamical astronomy

@article{Blanes2013NewFO, title={New families of symplectic splitting methods for numerical integration in dynamical astronomy}, author={S. Blanes and F. Casas and A. Farr{\'e}s and J. Laskar and J. Makazaga and A. Murua}, journal={Applied Numerical Mathematics}, year={2013}, volume={68}, pages={58-72} }

We present new splitting methods designed for the numerical integration of near-integrable Hamiltonian systems, and in particular for planetary N-body problems, when one is interested in very accurate results over a large time span. We derive in a systematic way an independent set of necessary and sufficient conditions to be satisfied by the coefficients of splitting methods to achieve a prescribed order of accuracy. Splitting methods satisfying such (generalized) order conditions are… Expand

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#### 68 Citations

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#### References

SHOWING 1-10 OF 32 REFERENCES

Splitting and composition methods in the numerical integration of differential equations

- Mathematics
- 2008

We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when… Expand

Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations

- Mathematics
- 2004

Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book.… Expand

Symplectic integrators and their application to dynamical astronomy

- Mathematics
- 1990

AbstractSymplectic integrators have many merits compared with traditional integrators:- the numerical solutions have a property of area preserving,- the discretization error in the energy integral… Expand

High order symplectic integrators for perturbed Hamiltonian systems

- Mathematics, Physics
- 2000

A family of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form H = A + εB was given in (McLachlan, 1995). We give here a constructive proof that for all… Expand

Symplectic maps for the N-body problem.

- Physics
- 1991

The present study generalizes the mapping method of Wisdom (1982) to encompass all gravitational n-body problems with a dominant central mass. The rationale for the generalized mapping method is… Expand

High precision symplectic integrators for the Solar System

- Mathematics, Physics
- 2013

Using a Newtonian model of the Solar System with all 8 planets, we perform extensive tests on various symplectic integrators of high orders, searching for the best splitting scheme for long term… Expand

MORE ON SYMPLECTIC CORRECTORS

- 1996

We give a different point of view of the symplectic correctors of Wisdom, Holman, and Touma and study the condition that errors committed by the kernel can be corrected. Because of this requirement… Expand

Numerical Hamiltonian Problems

- Mathematics
- 1994

Hamiltonian Systems. Examples of Hamiltonian Systems. Symplecticness. The solution operator. Preservation of area. Checking preservation of area: Jacobians. Checking preservation of area:… Expand

Processing Symplectic Methods for Near-Integrable Hamiltonian Systems

- Mathematics
- 2000

Processing techniques are used to approximate the exact flow of near-integrable Hamiltonian systems depending on a small perturbation parameter. We study the reduction of the number of conditions for… Expand

A long-term numerical solution for the insolation quantities of the Earth

- Physics
- 2004

We present here a new solution for the astronomical computation of the insolation quantities on Earth spanning from -250 Myr to 250 Myr. This solution has been improved with respect to La93 (Laskar… Expand