Geometric integration of ODEs using multiple quadratic auxiliary variables

TL;DR

This paper introduces a new geometric integrator for ODEs that preserves polynomial first integrals by using multiple quadratic auxiliary variables, unifying and extending existing midpoint and averaged vector field methods.

Contribution

The paper proposes a novel quadratic auxiliary variable approach to develop geometric integrators that preserve polynomial integrals, extending to higher-order methods via composition.

Findings

The method preserves polynomial first integrals in numerical solutions.

It generalizes the midpoint rule and averaged vector field method.

Numerical examples demonstrate improved conservation properties.

Abstract

We present a novel numerical method for solving ODEs while preserving polynomial first integrals. The method is based on introducing multiple quadratic auxiliary variables to reformulate the ODE as an equivalent but higher-dimensional ODE with only quadratic integrals to which the midpoint rule is applied. The quadratic auxiliary variables can subsequently be eliminated yielding a midpoint-like method on the original phase space. The resulting method is shown to be a novel discrete gradient method. Furthermore, the averaged vector field method can be obtained as a special case of the proposed method. The method can be extended to higher-order through composition and is illustrated through a number of numerical examples.