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

TL;DR

This paper develops structure-preserving discretization methods for boundary-controlled PDEs using port-Hamiltonian formalism, applying finite element techniques to hyperbolic and parabolic systems like wave and heat equations.

Contribution

It introduces a novel general structure for infinite-dimensional port-Hamiltonian systems enabling straightforward finite element discretization and applies this to multidimensional hyperbolic and parabolic PDEs.

Findings

Unified framework for hyperbolic and parabolic PDEs within port-Hamiltonian systems

Application of Partitioned Finite Element Method to these systems

Development of SCRIMP project for numerical simulation using FEniCS

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.