It is the purpose of this talk to analyze the nearly conservative behaviour of multi-value methods for the numerical solution of ordinary differential equations and, in particular, of Hamiltonian and oscillatory problems. Even if such methods cannot be symplectic, it is possible to lead them achieve a nearly preserving behavior through the properties of G-symplecticity [1,3], symmetry [1] and zero-growth parameters for the parasitic components [1]. We provide long-term error estimates for multi-value methods, in order to understand if such methods can really be assumed as good candidates for an excellent long-term invariants preservation. In particular, a backward error analysis is presented, which permits to get sharp estimates for the parasitic solution components and for the error in the Hamiltonian [2]. We also consider structure preservation properties in the numerical solution of oscillatory problems based on partial differential equations, tipically modelling oscillatory biological systems, whose solutions oscillate both in space and in time. Classical finite difference numerical methods for partial differential equations may not be well-suited to follow a prominent pe- riodic or oscillatory behaviour because, in order to accurately follow the oscillations, a very small stepsize would be required with corresponding deterioration of the numerical performances, especially in terms of efficiency. Special purpose numerical methods able to accurately retain the oscillatory behaviour are presented. [1] Raffaele D’Ambrosio, Giuseppe De Martino, Beatrice Paternoster, Numerical integration of Hamiltonian problems by G-symplectic methods, Adv. Comput. Math. 40(2), 553–575 (2014). [2] Raffaele D’Ambrosio, Ernst Hairer, Long-term stability of multi-value methods for ordinary differential equations, J. Sci. Comput., to appear. [3] Raffaele D’Ambrosio, Ernst Hairer, Christophe Zbinden, G-symplecticity implies conjugate-symplecticity of the underlying one-step method, BIT 53, 867–872 (2013).

Nearly preserving numerical methods for differential equations

D'AMBROSIO, RAFFAELE
2014

Abstract

It is the purpose of this talk to analyze the nearly conservative behaviour of multi-value methods for the numerical solution of ordinary differential equations and, in particular, of Hamiltonian and oscillatory problems. Even if such methods cannot be symplectic, it is possible to lead them achieve a nearly preserving behavior through the properties of G-symplecticity [1,3], symmetry [1] and zero-growth parameters for the parasitic components [1]. We provide long-term error estimates for multi-value methods, in order to understand if such methods can really be assumed as good candidates for an excellent long-term invariants preservation. In particular, a backward error analysis is presented, which permits to get sharp estimates for the parasitic solution components and for the error in the Hamiltonian [2]. We also consider structure preservation properties in the numerical solution of oscillatory problems based on partial differential equations, tipically modelling oscillatory biological systems, whose solutions oscillate both in space and in time. Classical finite difference numerical methods for partial differential equations may not be well-suited to follow a prominent pe- riodic or oscillatory behaviour because, in order to accurately follow the oscillations, a very small stepsize would be required with corresponding deterioration of the numerical performances, especially in terms of efficiency. Special purpose numerical methods able to accurately retain the oscillatory behaviour are presented. [1] Raffaele D’Ambrosio, Giuseppe De Martino, Beatrice Paternoster, Numerical integration of Hamiltonian problems by G-symplectic methods, Adv. Comput. Math. 40(2), 553–575 (2014). [2] Raffaele D’Ambrosio, Ernst Hairer, Long-term stability of multi-value methods for ordinary differential equations, J. Sci. Comput., to appear. [3] Raffaele D’Ambrosio, Ernst Hairer, Christophe Zbinden, G-symplecticity implies conjugate-symplecticity of the underlying one-step method, BIT 53, 867–872 (2013).
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/4416655
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