Conservative Methods for Dynamical Systems

Abstract

We show a novel systematic way to construct conservative finite difference\nschemes for quasilinear first-order system of ordinary differential equations\nwith conserved quantities. In particular, this includes both autonomous and\nnon-autonomous dynamical systems with conserved quantities of arbitrary forms,\nsuch as time-dependent conserved quantities. Sufficient conditions to construct\nconservative schemes of arbitrary order are derived using the multiplier\nmethod. General formulas for first-order conservative schemes are constructed\nusing divided difference calculus. New conservative schemes are found for\nvarious dynamical systems such as Euler's equation of rigid body rotation,\nLotka-Volterra systems, the planar restricted three-body problem and the damped\nharmonic oscillator.\n