Preserving Algebraic Invariants with Runge{kutta Methods

We study Runge{Kutta methods for the integration of ordinary diierential equations and their retention of algebraic invariants. As a general rule, we derive two conditions for the retention of such invariants. The rst is a condition on the co-eecients of the methods, the second is a pair of partial diierential equations that otherwise must be obeyed by the invariant. The cases related to the retention of quadratic and cubic invariant, 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 coeecients. 1 Background and notation In this paper we study the numerical solution by Runge{Kutta methods of the ordinary diierential system y 0 = f(t; y); y(0) = y 0 ; (1) for t 0, where y 2 R d and f : R + R d ! R d is a Lipschitz function. We assume that the exact solution y(t) of (1) is known to obey the condition that there exists a nontrivial function : R d R d ! R (or a family of such functions) such that (y(t); y 0) 0; t 0