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.

Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.