Functional equivariance and conservation laws in numerical integration

TL;DR

This paper introduces the concept of functional equivariance in numerical integrators, enabling the preservation of evolution laws of observables beyond invariants, with applications to PDEs and conservation laws.

Contribution

It develops a general framework for functional equivariance, linking invariant-preserving methods to those preserving local conservation laws in PDEs.

Findings

Integrators preserving quadratic invariants also preserve local conservation laws.

Symplectic integrators are shown to be multisymplectic.

The framework applies broadly to systems with important observable evolution laws.

Abstract

Preservation of linear and quadratic invariants by numerical integrators has been well studied. However, many systems have linear or quadratic observables that are not invariant, but which satisfy evolution equations expressing important properties of the system. For example, a time-evolution PDE may have an observable that satisfies a local conservation law, such as the multisymplectic conservation law for Hamiltonian PDEs. We introduce the concept of functional equivariance, a natural sense in which a numerical integrator may preserve the dynamics satisfied by certain classes of observables, whether or not they are invariant. After developing the general framework, we use it to obtain results on methods preserving local conservation laws in PDEs. In particular, integrators preserving quadratic invariants also preserve local conservation laws for quadratic observables, and symplectic integrators are multisymplectic.