Structure preserving discontinuous Galerkin approximation of one-dimensional port-Hamiltonian systems

TL;DR

This paper develops a structure-preserving discontinuous Galerkin method for discretizing one-dimensional port-Hamiltonian systems, ensuring energy conservation and stability, and demonstrates its effectiveness through simulations.

Contribution

It introduces a novel DG discretization that preserves the port-Hamiltonian structure and energy properties of 1D systems, including the effect of flux stabilization.

Findings

The method preserves power in conservative flux cases.

The spectrum is affected by the flux stabilization parameter.

Simulations validate the structure-preserving properties.

Abstract

In this article, we present the structure-preserving discretization of linear one-dimensional port-Hamiltonian (PH) systems of two conservation laws using discontinuous Galerkin (DG) methods. We recall the DG discretization procedure which is based on a subdivision of the computational domain, an elementwise weak formulation with up to two integration by parts, and the interconnection of the elements using several numerical fluxes. We present the interconnection of the element models, which is power preserving in the case of conservative (unstabilized) numerical fluxes, and we set up the resulting global PH state space model. We discuss the properties of the obtained models, including the effect of the flux stabilization parameter on the spectrum. Finally, we show simulations with different parameters for a boundary controlled linear hyperbolic system.