A Partitioned Finite Element Method for power-preserving discretization of open systems of conservation laws

Abstract

This paper presents a structure-preserving spatial discretization method for distributed parameter port-Hamiltonian systems. The class of considered systems are hyperbolic systems of two conservation laws in arbitrary spatial dimension and geometries. For these systems, a partioned finite element method is derived, based on the integration by parts of one of the two conservation laws written in weak form. The nonlinear 1D Shallow Water Equation (SWE) is first considered as a motivation example. Then the method is investigated on the example of the nonlinear 2D SWE. Complete derivation of the reduced finite-dimensional port-Hamiltonian system is provided and numerical experiments are performed. Extensions to curvilinear (polar) coordinate systems, space-varying coefficients and higher-order port-Hamiltonian systems (Euler-Bernoulli beam equation) are provided.

Stability and asymptotic analysis for instationary gas transport via relative energy estimates

“…(A2) Sufficiently smooth solutions (ρ, w) and (ρ, ŵ) to the system (1)-( 5) exist for parameters 0 ≤ ε, ε ≤ ε and 0 < γ ≤ γ, γ ≤ γ with γ, γ constant, which satisfy ρ ≤ ρ, ρ ≤ ρ and − w ≤ w, ŵ ≤ w. (8) Conditions (A1) and (A2) imply uniform bounds for P and its derivatives and ensure that the flow is subsonic; we refer to [6] for details. Under these conditions, we will show the following stability result: Let (ρ, w) and (ρ, ŵ) be sufficiently regular solutions of (1)- (3) with parameters ε, γ and ε, γ, and boundary values ĥ∂ and h ∂ as described in (6). Then…”

“…On the other hand, this formulation allows us to incorporate boundary conditions more naturally and to extend our results to networks in a straight-forward manner using appropriate coupling conditions at network junctions [24,9] that guarantee energy conservation or dissipation at network junctions. Similar formulations for compressible flow were also considered in the context of port-Hamiltonian systems; see [27,26] for the models and [3,19] for corresponding discretization strategies. Other systems, that fit into the general framework that we develop in this paper include the Euler-Korteweg system, the system of quantum hydrodynamics and the Euler-Poisson equations.…”

A port-Hamiltonian approach to modeling the structural dynamics of complex systems

“…A spatial discretization method is structure preserving if the resulting concentrated parameter system can be written in ISO form (1). We use PFEM, as proposed by Cardoso-Ribeiro et al [25], for reasons outlined in the introduction. In particular, due to its straightforward applicability and compatibility with standard FEM methods and software.…”

Numerical analysis of a structure-preserving space-discretization for an anisotropic and heterogeneous boundary controlled N-dimensional wave equation as port-Hamiltonian system

“…It seems that the Partitioned Finite Element Method (PFEM), first proposed in [12] and since then widely studied (see e.g. [8,9,54,13,11,53] and references therein), is one of the most adapted scheme to construct a mimetic finite-dimensional Dirac structure. Furthermore, it only relies on the well-proven and robust finite element method: it gives rise to sparse matrices, it is easy to implement, and one can take advantage of the numerous existing softwares.…”

“…Although PFEM was originally designed to mimic the Stokes-Dirac structure at the discrete level in [12] using vectorial calculus, it has also been rewritten in the formalism of exterior calculus in [13], which usually allows easier proofs of the aforementioned cohomology and topological decomposition preservations. It will be shown that together with the compatibility conditions between the spaces of approximations coming from cohomoly, PFEM leads to a very interesting result for the space discretization of pHs, given in Theorem 4.4.…”