Discretization of the Burgers' equation as a port-Hamiltonian system

TL;DR

This paper develops port-Hamiltonian formulations for the Burgers' equation, enabling stable numerical simulation with boundary control, by incorporating viscous effects and analyzing stability conditions.

Contribution

It introduces a novel port-Hamiltonian framework for both inviscid and viscous Burgers' equations, facilitating boundary control and stable discretization.

Findings

Port-Hamiltonian formulation effectively models convective and dissipative effects.

Finite element discretization yields a finite-dimensional port-Hamiltonian system.

Numerical experiments confirm the approach's stability and accuracy.

Abstract

The numerical simulation of the inviscid Burgers' equation is often hindered by spurious oscillations near discontinuities. To mitigate this issue, a viscous term can be introduced, leading to the viscous Burgers' equation. In this work, port-Hamiltonian formulations for both the inviscid and the viscous Burgers' equations are proposed, enabling a representation that incorporates both convective and dissipative effects. Boundary control and observation are naturally handled within this framework. Applying a dedicated finite element method, a finite-dimensional port-Hamiltonian system is derived. The relationship between time step, spatial resolution, and viscosity required to achieve numerical stability is analyzed. Numerical experiments validate the effectiveness of the approach.