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Numerical integration of Hamiltonian problems by G-symplectic methods

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, University of Salerno, via Giovanni Paolo II, 132 - 84084, Fisciano, (Sa), Italy

    Raffaele D’Ambrosio, Giuseppe De Martino & Beatrice Paternoster

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  1. Raffaele D’Ambrosio
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  2. Giuseppe De Martino
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  3. Beatrice Paternoster
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Correspondence to Raffaele D’Ambrosio.

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Communicated by: Kendall Atkinson

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D’Ambrosio, R., De Martino, G. & Paternoster, B. Numerical integration of Hamiltonian problems by G-symplectic methods. Adv Comput Math 40, 553–575 (2014). https://doi.org/10.1007/s10444-013-9318-z

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  • DOI: https://doi.org/10.1007/s10444-013-9318-z

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