Preserving algebraic invariants with Runge–Kutta methods

We study Runge–Kutta methods for the integration of ordinary differential equations and the retention of algebraic invariants. As a general rule, we derive two conditions for the retention of such invariants. The first is a condition on the coefficients of the methods, the second is a pair of partial differential equations that otherwise must be obeyed by the invariant. This paper extends previous work on multistep methods in Iserles (Technical Report NA1997/13, DAMTP, University of Cambridge, 1997). The cases related to the retention of quadratic and cubic invariants, perhaps of greatest relevance in applications, are thoroughly discussed. We conclude recommending a generalized class of Runge–Kutta schemes, namely Lie-group-type Runge–Kutta methods. These are schemes for the solution of ODEs on Lie groups but can be employed, together with group actions, to preserve a larger class of algebraic invariants without restrictions on the coefficients.