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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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Quadratic pseudospectrum for identifying localized states
We examine the utility of the quadratic pseudospectrum for understanding and detecting states that are somewhat localized in position and energy, in particular, in the context of condensed matter physics. Specifically, the quadratic pseudospectrum represents a method for approaching systems with incompatible observables { A j ∣1 ≤ j ≤ d} as it minimizes collectively the errors ‖ A j v − λ j v‖ while defining a joint approximate spectrum of incompatible observables. Moreover, we derive an important estimate relating the Clifford and quadratic pseudospectra. Finally, we prove that the quadratic pseudospectrum is local and derive the bounds on the errors that are incurred by truncating the system in the vicinity of where the pseudospectrum is being calculated.
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- Award ID(s):
- 2110398
- PAR ID:
- 10394905
- Date Published:
- Journal Name:
- Journal of Mathematical Physics
- Volume:
- 64
- Issue:
- 2
- ISSN:
- 0022-2488
- Page Range / eLocation ID:
- 023501
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0098336
