2020/07/16 by Andrea Brugnoli, Ghislain Haine, Brugnoli, Andrea +5
Engineering · Mathematics · Physics and Astronomy · #35K90 #35L90 #65M60 #Advanced Numerical Methods in Computational Mathematics #Dynamical Systems (math.DS) #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2007.08326
openalex publication_date 2020/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the design of structure-preserving discretization methods for the\nsolution of systems of boundary controlled Partial Differential Equations\n(PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel\ngeneral structure of infinite-dimensional port-Hamiltonian systems (pHs) for\nwhich the Partitioned Finite Element Method (PFEM) straightforwardly applies.\nThe proposed strategy is applied to abstract multidimensional linear hyperbolic\nand parabolic systems of PDEs. Then we show that instructional model problems\nbased on the wave equation, Mindlin equation and heat equation fit within this\nunified framework. Secondly we introduce the ongoing project SCRIMP (Simulation\nand ContRol of Interactions in Multi-Physics) developed for the numerical\nsimulation of infinite-dimensional pHs. SCRIMP notably relies on the FEniCS\nopen-source computing platform for the finite element spatial discretization.\nFinally, we illustrate how to solve the considered model problems within this\nframework by carefully explaining the methodology. As additional support,\ncompanion interactive Jupyter notebooks are available.\n