On the preservation of second integrals by 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 leverage the recently discovered theory of discrete second integrals to show novel results about Runge-Kutta methods. In particular, we show that any Runge-Kutta method preserves all affine second integrals but cannot preserve all quadratic second integrals of an ODE. A number of interesting corollaries are also discussed, such as the preservation of certain rational integrals by Runge-Kutta methods. The special case of affine second integrals with constant cofactor are also discussed as well the preservation of third and higher integrals.