Structure-preserving reduced basis methods for Poisson systems
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- by Jan S. Hesthaven and Cecilia Pagliantini;
- Math. Comp. 90 (2021), 1701-1740
- DOI: https://doi.org/10.1090/mcom/3618
- Published electronically: April 30, 2021
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Abstract:
We develop structure-preserving reduced basis methods for a large class of nondissipative problems by resorting to their formulation as Hamiltonian dynamical systems. With this perspective, the phase space is naturally endowed with a Poisson manifold structure which encodes the physical properties, symmetries, and conservation laws of the dynamics. The goal is to design reduced basis methods for the general state-dependent degenerate Poisson structure based on a two-step approach. First, via a local approximation of the Poisson tensor, we split the Hamiltonian dynamics into an “almost symplectic” part and the trivial evolution of the Casimir invariants. Second, canonically symplectic reduced basis techniques are applied to the nontrivial component of the dynamics, preserving the local Poisson tensor kernel exactly. The global Poisson structure and the conservation properties of the phase flow are retained by the reduced model in the constant-valued case and up to errors in the Poisson tensor approximation in the state-dependent case. A priori error estimates for the solution of the reduced system are established. A set of numerical simulations is presented to corroborate the theoretical findings.References
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Bibliographic Information
- Jan S. Hesthaven
- Affiliation: MCSS, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
- MR Author ID: 350602
- ORCID: 0000-0001-8074-1586
- Email: jan.hesthaven@epfl.ch
- Cecilia Pagliantini
- Affiliation: CASA, Eindhoven University of Technology (TU/e), Eindhoven, The Netherlands
- MR Author ID: 1150107
- Email: c.pagliantini@tue.nl
- Received by editor(s): February 28, 2020
- Received by editor(s) in revised form: August 31, 2020, and November 9, 2020
- Published electronically: April 30, 2021
- Additional Notes: This work was partially supported by AFOSR under grant FA9550-17-1-9241.
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 1701-1740
- MSC (2020): Primary 15A21, 53D17, 53D22, 37N30, 37J35, 78M34, 65P10
- DOI: https://doi.org/10.1090/mcom/3618
- MathSciNet review: 4273113