Functional equivariance and modified vector fields

TL;DR

This paper analyzes how structure-preserving numerical integrators affect the evolution of observables in systems with functional equivariance, extending backward error analysis to include invariant and symplectic properties.

Contribution

It provides a characterization of the evolution of observables via modified vector fields, generalizing results on invariant preservation and symplecticity in the context of functional equivariance.

Findings

Modified vector fields describe the evolution of affine and quadratic observables.

Results on invariant preservation are extended to non-invariant observables.

Symplecticity properties of modified vector fields are generalized.

Abstract

This paper examines functional equivariance, recently introduced by McLachlan and Stern [Found. Comput. Math. (2022)], from the perspective of backward error analysis. We characterize the evolution of certain classes of observables (especially affine and quadratic) by structure-preserving numerical integrators in terms of their modified vector fields. Several results on invariant preservation and symplecticity of modified vector fields are thereby generalized to describe the numerical evolution of non-invariant observables.