It is the purpose of this talk to analyze the employ of General Linear Methods (GLMs) for the numerical integration of Hamiltonian problems. Indeed, even if the numerical flow generated by a GLM cannot be symplectic, Butcher recently introduced in [1] a concept of near conservation, denoted as G-symplecticity, for such methods which, properly combined with other desirable features (such as symmetry and boundedness of parasitic components), allows to achieve a very accurate long time conservation of the Hamiltonian. We also focus our attention on the connections between the order of convergence of a GLM and the observable Hamiltonian deviation, by employing the theory of B-series [3]. Moreover, we derive a semi-implicit GLM [2] which results competitive with respect to symplectic Runge-Kutta methods. Numerical results on a selection of Hamiltonian problems are presented, confirming the structure-preserving capability of G-symplectic integrators. References [1] J.C. Butcher, Numerical methods for Ordinary Differential Equations, Second Edition, Wiley, Chichester, 2008. [2] R. D’Ambrosio, G. De Martino and B. Paternoster, Construction of nearly conservative multivalue numerical methods for Hamiltonian problems, Comm. Appl. Ind. Math, doi: 10.1685/journal.caim.412 (2013). [3] R. D’Ambrosio, G. De Martino and B. Paternoster, Numerical integration of Hamiltonian problems by G-symplectic integrators, submitted.
Titolo: | Numerical solution of Hamiltonian problems by G-symplectic integrators |
Autori: | |
Data di pubblicazione: | 2013 |
Abstract: | It is the purpose of this talk to analyze the employ of General Linear Methods (GLMs) for the numerical integration of Hamiltonian problems. Indeed, even if the numerical flow generated by a GLM cannot be symplectic, Butcher recently introduced in [1] a concept of near conservation, denoted as G-symplecticity, for such methods which, properly combined with other desirable features (such as symmetry and boundedness of parasitic components), allows to achieve a very accurate long time conservation of the Hamiltonian. We also focus our attention on the connections between the order of convergence of a GLM and the observable Hamiltonian deviation, by employing the theory of B-series [3]. Moreover, we derive a semi-implicit GLM [2] which results competitive with respect to symplectic Runge-Kutta methods. Numerical results on a selection of Hamiltonian problems are presented, confirming the structure-preserving capability of G-symplectic integrators. References [1] J.C. Butcher, Numerical methods for Ordinary Differential Equations, Second Edition, Wiley, Chichester, 2008. [2] R. D’Ambrosio, G. De Martino and B. Paternoster, Construction of nearly conservative multivalue numerical methods for Hamiltonian problems, Comm. Appl. Ind. Math, doi: 10.1685/journal.caim.412 (2013). [3] R. D’Ambrosio, G. De Martino and B. Paternoster, Numerical integration of Hamiltonian problems by G-symplectic integrators, submitted. |
Handle: | http://hdl.handle.net/11386/4416662 |
Appare nelle tipologie: | 4.2 Abstract |