Numerical Approximation of Port-Hamiltonian Systems for Hyperbolic or Parabolic PDEs with Boundary Control

Abstract

We consider the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel general structure of infinite-dimensional port-Hamiltonian systems (pHs) for which the Partitioned Finite Element Method (PFEM) straightforwardly applies. The proposed strategy is applied to abstract multidimensional linear hyperbolic and parabolic systems of PDEs. Then we show that instructional model problems based on the wave equation, Mindlin equation and heat equation fit within this unified framework. Secondly, we introduce the ongoing project SCRIMP (Simulation and Control of Interactions in Multi-Physics) developed for the numerical simulation of infinite-dimensional pHs. SCRIMP notably relies on the FEniCS open-source computing platform for the finite element spatial discretization. Finally, we illustrate how to solve the considered model problems within this framework by carefully explaining the methodology. As additional support, companion interactive Jupyter notebooks are available.

Numerical approximation of port-Hamiltonian systems for hyperbolic or parabolic PDEs with boundary control

“…Taking advantage of the strong underlying structure, we finally describe a unified object-oriented implementation of these models via PFEM. Companion interactive Jupyter notebooks [16] are discussed to illustrate our methodology.…”

“…In Section 4 the ongoing environment SCRIMP is described in detail. In Section 5 the three companion interactive Jupyter notebooks [16] are thoroughly explained. Conclusions and perspectives are finally drawn in Section 6.…”

“…The aim is to provide a flexible prototype Python code for the numerical simulation of pHs both for research and educational purposes. In addition to numerical experiments proposed later in Section 5, the reader is referred to interactive companion Jupyter notebooks [16] to learn how to numerically solve the model problems introduced in Section 3 with SCRIMP. In the following, the key ideas behind SCRIMP are mentioned and then a specific emphasis on both space and time discretizations is given.…”

“…The design of SCRIMP is based on procedural and object-oriented paradigms and thus follows the standard ideas governing most of the numerical PDE software. Whereas a detailed exposition of the design patterns of SCRIMP and its performance will be published elsewhere, concrete illustrations of most of these key ideas can be found in the companion Jupyter notebooks [16]. The description of the current numerical methods related to space and time discretizations available in SCRIMP is given.…”

“…In particular, the structure-preserving model reduction algorithm (Algorithm 1) proposed in [20] has been selected in the pHODE case. We refer the reader to [16] for an illustration, where the model reduction of the pHs related to the wave equation problem is considered. While for linear pHDAE systems consolidated methodologies have been proposed (see, e.g., [23]), structure-preserving model reduction for general nonlinear differential algebraic systems remains to be explored, to the best of our knowledge.…”