Discretization of the Burgers' equation as a port-Hamiltonian system
This work addresses numerical stability issues in simulating the Burgers' equation, which is incremental for computational fluid dynamics and control theory applications.
The authors tackled the problem of spurious oscillations in numerical simulations of the Burgers' equation by proposing port-Hamiltonian formulations for both inviscid and viscous versions, enabling a representation that handles convective and dissipative effects, with numerical experiments validating the approach.
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.