Numerical integration of Hamiltonian problems by G-symplectic methods

It is the purpose of this paper to consider the employ of General Linear Methods (GLMs) as geometric numerical solvers for the treatment of Hamiltonian problems. Indeed, even if the numerical flow generated by a GLM cannot be symplectic, we exploit here a concept of near conservation for such method...

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Bibliographic Details
Published in:Advances in computational mathematics 2014-04, Vol.40 (2), p.553-575
Main Authors: D’Ambrosio, Raffaele, De Martino, Giuseppe, Paternoster, Beatrice
Format: Article
Language:English
Subjects:
Computational Mathematics and Numerical Analysis
Computational Science and Engineering
Conservation
Convergence
Deviation
Joints
Mathematical and Computational Biology
Mathematical Modeling and Industrial Mathematics
Mathematical models
Mathematics
Mathematics and Statistics
Runge-Kutta method
Solvers
Symmetry
Visualization
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Summary:It is the purpose of this paper to consider the employ of General Linear Methods (GLMs) as geometric numerical solvers for the treatment of Hamiltonian problems. Indeed, even if the numerical flow generated by a GLM cannot be symplectic, we exploit here a concept of near conservation for such methods which, properly combined with other desirable features (such as symmetry and boundedness of parasitic components), allows to achieve an accurate conservation of the Hamiltonian. In this paper we focus our attention on the connection between order of convergence and Hamiltonian deviation by multivalue methods. Moreover, we derive a semi-implicit GLM which results competitive to symplectic Runge-Kutta methods, which are notoriously implicit.
ISSN:1019-7168
1572-9044
DOI:10.1007/s10444-013-9318-z