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Foundations of Computational Mathematics
Published in:

20-09-2022

Functional Equivariance and Conservation Laws in Numerical Integration

Authors: Robert I. McLachlan, Ari Stern

Published in: Foundations of Computational Mathematics | Issue 1/2024

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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.
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Footnotes
1
For some methods, such as implicit Runge–Kutta methods, \( \Phi _{ \Delta t , f } (y) \) might only be defined for sufficiently small \( \Delta t \). Including such integrators requires only the minor modification of viewing \( \Phi _f \) as a partial function.
 
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Metadata
Title
Functional Equivariance and Conservation Laws in Numerical Integration
Authors
Robert I. McLachlan
Ari Stern
Publication date
20-09-2022
Publisher
Springer US
Published in
Foundations of Computational Mathematics / Issue 1/2024
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-022-09590-8

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