Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems
For researchers in geometric discretization of port-Hamiltonian systems, this work provides a novel method for structure-preserving discretization of hyperbolic conservation laws.
The paper develops a mixed Galerkin discretization for distributed parameter port-Hamiltonian systems, preserving the Stokes-Dirac structure and enabling structure-preserving schemes with trade-offs between centered and upwind approximations. The method is illustrated on Whitney finite elements in 2D and compared with a related approach in 1D.
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.