Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems

TL;DR

This paper develops a geometric discretization method for port-Hamiltonian systems using a mixed Galerkin approach, enabling structure-preserving finite-dimensional models that balance accuracy and stability.

Contribution

It introduces a weak formulation of Stokes-Dirac structures and a finite-dimensional approximation via mixed Galerkin methods with power-preserving maps, advancing geometric discretization techniques.

Findings

The method accurately approximates eigenvalues in 1D.

It provides a flexible framework balancing centered and upwind schemes.

Finite element implementation demonstrated on 2D triangulations.

Abstract

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.