Conservative methods for dynamical systems

TL;DR

This paper introduces a systematic approach to constructing conservative finite difference schemes for quasilinear first-order ODE systems, ensuring preservation of conserved quantities including time-dependent ones.

Contribution

It develops a general framework using the multiplier method and divided difference calculus to create high-order conservative schemes for diverse dynamical systems.

Findings

New conservative schemes for Euler's rigid body rotation

Conservative methods for Lotka-Volterra systems

Schemes for the three-body problem and harmonic oscillator

Abstract

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.