Symplectic Methods Based on Decompositions

Abstract

We consider methods that integrate systems of differential equations dy/dt = f (y) by taking advantage of a decomposition of the right-hand side f = f [ν]. We derive a general necessary and sufficient condition for those methods to be symplectic for Hamiltonian problems. Special attention is given to the case of additive Runge-Kutta methods.

A note on B-stability of splitting methods

“…We may easily prove that these two conditions imply that b [1] 1 = 1 2 and that a [1] 11 = a [1] 33 = c, with c a free parameter such that c ≥ 1 4 . So, the class of FSRK methods (7) Using composition methods, we may describe a procedure that enables us to construct higher order B-stable numerical methods.…”

“…If, in (1), N = 2 and f [1] is stiff while f [2] is not, then it is common to combine an implicit integrator for f [1] with an explicit integrator for f [2] . For instance, in a reactiondiffusion partial differential problem we may combine an implicit method for the diffusion with an explicit one for the reaction [2].…”

“…where α is a real constant and ψ MP h,g denotes a step of length h of the implicit midpoint rule applied to a differential system with right-hand side g. Clearly (2) is a multiple time-step method that uses f [2] less frequently than f [1] .…”

“…Without loss of generality we may consider θ = [1, 2, 1] T . The order conditions are the following [1].…”

“…Thus, in a FSRK method (7) of order 2 with s = 3 and θ = [1, 2, 1] T , the coefficients a [1] 11 , a [1] 33 and b [1] 1 are free parameters.…”