Invariantization of numerical schemes using moving frames

Abstract

This paper deals with a geometric technique to construct numerical schemes for differential equations that inherit Lie symmetries. The moving frame method enables one to adjust existing numerical schemes in a geometric manner and systematically construct proper invariant versions of them. Invariantization works as an adaptive transformation on numerical solutions, improving their accuracy greatly. Error reduction in the Runge-Kutta method by invariantization is studied through several applications including a harmonic oscillator and a Hamiltonian system.