Journal
JOURNAL OF COMPUTATIONAL PHYSICS
Volume 361, Issue -, Pages 442-476Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2018.02.006
Keywords
Systems of conservation laws with boundary energy flows; Port-Hamiltonian systems; Mixed Galerkin methods; Geometric spatial discretization; Structure-preserving discretization
Funding
- European Union's Horizon research and innovation programme (Marie Sklodowska-Curie Individual Fellowship) [655204]
- Agence Nationale de la Recherche, (ANR) project INFIDHEM [ANR-16-CE92-0028]
- Marie Curie Actions (MSCA) [655204] Funding Source: Marie Curie Actions (MSCA)
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We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes-Dirac) structure with a segmented boundary according to the causality of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac structure by a finite-dimensional Dirac structure is realized using a mixed Galerkin approach and power-preserving linear maps, which define minimal discrete power variables. (iii) With a consistent approximation of the Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models. By the degrees of freedom in the power-preserving maps, the resulting family of structure-preserving schemes allows for trade-offs between centered approximations and upwinding. We illustrate the method on the example of Whitney finite elements on a 2D simplicial triangulation and compare the eigenvalue approximation in 1D with a related approach. (C) 2018 Elsevier Inc. All rights reserved.
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