On the preservation of second integrals by Runge-Kutta methods

TL;DR

This paper investigates how Runge-Kutta methods preserve second integrals of ODEs, showing they maintain affine second integrals but generally not quadratic ones, with implications for numerical analysis of integrability.

Contribution

It provides a theoretical analysis of the preservation of second integrals by Runge-Kutta methods, highlighting which types are preserved and under what conditions.

Findings

Affine second integrals are preserved by all Runge-Kutta methods.

Quadratic second integrals are generally not preserved.

Certain rational integrals are preserved by arbitrary Runge-Kutta methods.

Abstract

One can elucidate integrability properties of ordinary differential equations (ODEs) by knowing the existence of second integrals (also known as weak integrals or Darboux polynomials for polynomial ODEs). However, little is known about how they are preserved, if at all, under numerical methods. Here, we show that in general all Runge-Kutta methods will preserve all affine second integrals but not (irreducible) quadratic second integrals. A number of interesting corollaries are discussed, such as the preservation of certain rational integrals by arbitrary Runge-Kutta methods. We also study the special case of affine second integrals with constant cofactor and discuss the preservation of affine higher integrals.