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

TL;DR

This paper explores linear projection methods for solving autonomous ODEs that preserve first integrals, demonstrating their equivalence to discrete gradient methods and establishing conditions for their existence, uniqueness, and accuracy.

Contribution

It establishes the theoretical equivalence between projection methods and discrete gradient methods, extending existing results to multiple integrals and providing a unified framework.

Findings

Projection methods preserve first integrals under mild conditions.

Existence and local uniqueness of solutions are proven for single integral cases.

Numerical examples support the theoretical results for multiple integrals.

Abstract

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.