Conservative methods for dynamical systems
It provides a general framework for preserving conservation laws in numerical simulations, which is important for accurate long-term integration of dynamical systems.
The paper presents a systematic method for constructing conservative finite difference schemes for dynamical systems with conserved quantities, including time-dependent ones, and demonstrates first-order schemes for several systems like Euler's rigid body rotation and Lotka-Volterra.
We show a novel systematic way to construct conservative finite difference schemes for quasilinear first-order system of ordinary differential equations with conserved quantities. In particular, this includes both autonomous and non-autonomous dynamical systems with conserved quantities of arbitrary forms, such as time-dependent conserved quantities. Sufficient conditions to construct conservative schemes of arbitrary order are derived using the multiplier method. General formulas for first-order conservative schemes are constructed using divided difference calculus. New conservative schemes are found for various dynamical systems such as Euler's equation of rigid body rotation, Lotka-Volterra systems, the planar restricted three-body problem and the damped harmonic oscillator.