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Functional Equivariance and Conservation Laws in Numerical Integration

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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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Notes

  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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Acknowledgements

The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program “Geometry, compatibility and structure preservation in computational differential equations,” when work on this paper was undertaken. This program was supported by EPSRC grant number EP/R014604/1. Robert McLachlan was supported in part by the Marsden Fund of the Royal Society of New Zealand and by a fellowship from the Simons Foundation. Ari Stern was supported in part by NSF grant DMS-1913272.

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Authors and Affiliations

  1. School of Fundamental Sciences, Massey University, Palmerston North, New Zealand

    Robert I. McLachlan

  2. Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, USA

    Ari Stern

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  1. Robert I. McLachlan
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Correspondence to Ari Stern.

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Communicated by Arieh Iserles.

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McLachlan, R.I., Stern, A. Functional Equivariance and Conservation Laws in Numerical Integration. Found Comput Math 24, 149–177 (2024). https://doi.org/10.1007/s10208-022-09590-8

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