Preserving multiple first integrals by discrete gradients

TL;DR

This paper introduces integrators that exactly preserve multiple first integrals of ODE systems by modifying existing methods through projection and local coordinate approaches, demonstrated on the Kepler problem.

Contribution

It presents a novel framework for constructing integrators that conserve multiple first integrals simultaneously using discrete gradients and tangent space concepts.

Findings

Methods successfully conserve all first integrals in numerical tests

Approaches are applicable to standard schemes like Runge-Kutta

Validated on the Kepler problem with positive results

Abstract

We consider systems of ordinary differential equations with known first integrals. The notion of a discrete tangent space is introduced as the orthogonal complement of an arbitrary set of discrete gradients. Integrators which exactly conserve all the first integrals simultaneously are then defined. In both cases we start from an arbitrary method of a prescribed order (say, a Runge-Kutta scheme) and modify it using two approaches: one based on projection and one based one local coordinates. The methods are tested on the Kepler problem.