Projection methods and discrete gradient methods for preserving first integrals of ODEs

In this paper we study linear projection methods for approximating the solution and simultaneously preserving first integrals of autonomous ordinary differential equations. We show that (linear) projection methods are a subset of discrete gradient methods. In particular, each projection method is equivalent to a class of discrete gradient methods (where the choice of discrete gradient is arbitrary) and earlier results for discrete gradient methods also apply to projection methods. Thus we prove that for the case of preserving one first integral, under certain mild conditions, the numerical solution for a projection method exists and is locally unique, and preserves the order of accuracy of the underlying method. In the case of preserving multiple first integrals the relationship between projection methods and discrete gradient methods persists. Moreover, numerical examples show that similar existence and order results should also hold for the multiple integral case. For completeness we show how existing projection methods from the literature fit into our general framework.