It is the purpose of this talk to analyze the structure preservation properties of multi-value methods for the numerical solution of Hamiltonian problems, originating from Celestial Mechanics, Molecular Dynamics and Immunology. In particular, we aim to achieve accurate and ecient numerical energy preservation and orbits computation in the dynamics of Solar system planets, by employing real data desumed by Nasa Horizons System, as well as numerical modeling of T-cell dynamics by discretization of suitable models arising from Mechanical Statistics is object of the investigations. It is known that, in the spirit of numerical conservation of the invariants of Hamiltonian problems, the classical symplecticity property play a crucial role. However, only certain Runge-Kutta methods are candidate for symplecticity. Even if multivalue methods cannot be symplectic, it is possible to lead them possess a computationally cheap nearly preserving behavior through the properties of G-symplecticity, symmetry and zero-growth parameters for the parasitic components. We are particularly interested in the long-time behavior of multi-value methods. Hence, we provide long-term error estimates by backward error analysis arguments, which permits to get sharp estimates for the parasitic solution components and for the error in the Hamiltonian. We prove that the eects of parasitism on the numerical solution are then negligible on time intervals of length O(h2), where h is the stepsize of integration. The theoretical expectations are then conrmed by the numerical evidence. References: 1. R. D'Ambrosio, G. De Martino, B. Paternoster, Numerical integration of Hamiltonian problems by G-symplectic methods, Adv. Comput. Math. 40(2), 553-575 (2014). 2. R. D'Ambrosio, E. Hairer, Long-term stability of multi-value methods for ordinary dierential equations, J. Sci. Comput. doi: 10.1007/s10915-013-9812-y (2013). 3. R. D'Ambrosio, E. Hairer, C. Zbinden, G-symplecticity implies conjugate-symplecticity of the underlying one-step method. BIT vol. 53, 867-872 (2013).

Long-term structure-preserving numerical methods for Hamiltonian problems in Physics and Medicine

D'AMBROSIO, RAFFAELE
2014

Abstract

It is the purpose of this talk to analyze the structure preservation properties of multi-value methods for the numerical solution of Hamiltonian problems, originating from Celestial Mechanics, Molecular Dynamics and Immunology. In particular, we aim to achieve accurate and ecient numerical energy preservation and orbits computation in the dynamics of Solar system planets, by employing real data desumed by Nasa Horizons System, as well as numerical modeling of T-cell dynamics by discretization of suitable models arising from Mechanical Statistics is object of the investigations. It is known that, in the spirit of numerical conservation of the invariants of Hamiltonian problems, the classical symplecticity property play a crucial role. However, only certain Runge-Kutta methods are candidate for symplecticity. Even if multivalue methods cannot be symplectic, it is possible to lead them possess a computationally cheap nearly preserving behavior through the properties of G-symplecticity, symmetry and zero-growth parameters for the parasitic components. We are particularly interested in the long-time behavior of multi-value methods. Hence, we provide long-term error estimates by backward error analysis arguments, which permits to get sharp estimates for the parasitic solution components and for the error in the Hamiltonian. We prove that the eects of parasitism on the numerical solution are then negligible on time intervals of length O(h2), where h is the stepsize of integration. The theoretical expectations are then conrmed by the numerical evidence. References: 1. R. D'Ambrosio, G. De Martino, B. Paternoster, Numerical integration of Hamiltonian problems by G-symplectic methods, Adv. Comput. Math. 40(2), 553-575 (2014). 2. R. D'Ambrosio, E. Hairer, Long-term stability of multi-value methods for ordinary dierential equations, J. Sci. Comput. doi: 10.1007/s10915-013-9812-y (2013). 3. R. D'Ambrosio, E. Hairer, C. Zbinden, G-symplecticity implies conjugate-symplecticity of the underlying one-step method. BIT vol. 53, 867-872 (2013).
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/4416657
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