This talk is devoted to the investigation of the canonical properties of general linear methods for long time integration of Hamiltonian problems. Runge-Kutta methods exhibit some fundamental canonical properties if they are symplectic. It is known that general linear methods cannot be symplectic (see [3]), however it is possible to inherit from their nonlinear stability properties a nearly canonical behavior, known as G-symplecticity [1,2], which is the first ingredient to obtain an accurate conservation of the invariants of an Hamiltonian problem. Due to their multivalue nature, general linear methods generates a parasitic behavior of the numerical solution which needs to be properly removed: we discuss how G-symplectic general linear methods free from parasitism can be developed. The third aspect we aim to discuss is symmetry: in particular, we explain how time reversal symmetry allows to derive methods of a certain order by applying a reduced number of order conditions. Numerical experiments on a selection of Hamiltonian problems are discussed. 1. J. C. Butcher 2008 Numerical methods for Ordinary Differential Equations, Second Edition, Wiley. 2. J. C. Butcher, R. D’Ambrosio, Partitioned general linear methods for separable Hamiltonian problems, in preparation. 3. J. C. Butcher and L. L. Hewitt 2009 The existence of symplectic general linear methods, Numer. Algor. 51, 77-84.

Canonical Properties of General Linear Methods for Hamiltonian Problems

D'AMBROSIO, RAFFAELE
2012

Abstract

This talk is devoted to the investigation of the canonical properties of general linear methods for long time integration of Hamiltonian problems. Runge-Kutta methods exhibit some fundamental canonical properties if they are symplectic. It is known that general linear methods cannot be symplectic (see [3]), however it is possible to inherit from their nonlinear stability properties a nearly canonical behavior, known as G-symplecticity [1,2], which is the first ingredient to obtain an accurate conservation of the invariants of an Hamiltonian problem. Due to their multivalue nature, general linear methods generates a parasitic behavior of the numerical solution which needs to be properly removed: we discuss how G-symplectic general linear methods free from parasitism can be developed. The third aspect we aim to discuss is symmetry: in particular, we explain how time reversal symmetry allows to derive methods of a certain order by applying a reduced number of order conditions. Numerical experiments on a selection of Hamiltonian problems are discussed. 1. J. C. Butcher 2008 Numerical methods for Ordinary Differential Equations, Second Edition, Wiley. 2. J. C. Butcher, R. D’Ambrosio, Partitioned general linear methods for separable Hamiltonian problems, in preparation. 3. J. C. Butcher and L. L. Hewitt 2009 The existence of symplectic general linear methods, Numer. Algor. 51, 77-84.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/4419060
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