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, 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.