Nearly conservative multivalue methods for separable Hamiltonian problems

This talk investigates the canonical properties of general linear methods for long time integration of Hamiltonian problems. It is known that the classical symplecticity property is important for the accurate numerical solution of Hamiltonian problems and this is only possible for "canonical" Runge-Kutta methods. Even if general linear methods cannot be symplectic (see [3]), it is possible to lead them inherit a nearly canonical behavior from their nonlinear stability properties. This is done by imposing a further algebraic constraint on their coecient matrices, known as G-symplecticity [1], which is a rst requirement to obtain an accurate conservation of the invariants of an Hamiltonian problem. Special attention will be given to the numerical treatment of separable Hamiltonian problems: to this purpose, the family of G-symplectic partitioned general linear methods is introduced [2]. Due to their multivalue nature, partitioned general linear methods generate parasitic components of the numerical solution which needs to be properly removed: we discuss how G-symplectic partitioned general linear methods free from parasitism can be constructed. We also consider the eects of G-symplecticity on the order of convergence of the derived methods, by exploiting the theory of B-series. Numerical experiments on a selection of separable Hamiltonian problems are discussed. This is a joint work with J. C. Butcher from the University of Auckland (New Zealand). [1] J. C. Butcher 2008 Numerical methods for Ordinary Dierential 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