Energy-consistent Petrov-Galerkin time discretization of port-Hamiltonian systems

TL;DR

This work develops a high-order, energy-consistent continuous Petrov–Galerkin time discretization for nonlinear port-Hamiltonian systems, applicable to both infinite- and finite-dimensional models and compatible with structure-preserving space discretizations. A key innovation is the $L^2$-projection in time, combined with Gauss quadrature, which yields energy-consistency at time grid points for general Hamiltonians and nonlinear operators, while reducing to classical cPG in the quadratic-linear case. Numerical experiments on the Toda lattice, spinning rigid body, quasilinear wave equation, and porous medium equation demonstrate optimal time-convergence rates, nodal superconvergence, and energy-balance accuracy up to machine precision provided the projection and quadrature are suitably chosen. The framework offers a robust, high-order approach to preserving port-Hamiltonian structure in both time and space for a broad class of nonlinear evolution equations and DAEs.

Abstract

For a general class of nonlinear port-Hamiltonian systems we develop a high-order time discretization scheme with certain structure preservation properties. The finite or infinite-dimensional system under consideration possesses a Hamiltonian function, which represents an energy in the system and is conserved or dissipated along solutions. For infinite-dimensional systems this structure is preserved under suitable Galerkin discretization in space. The numerical scheme is energy-consistent in the sense that the Hamiltonian of the approximate solutions at time grid points behaves accordingly. This structure preservation property is achieved by specific design of a continuous Petrov-Galerkin (cPG) method in time. It coincides with standard cPG methods in special cases, in which the latter are energy-consistent. Examples of port-Hamiltonian ODEs and PDEs are presented to visualize the framework. In numerical experiments the energy consistency is verified and the convergence behavior is investigated.

Figures (9)

• Figure 1: Convergence in $\tau$ for the Toda lattice \ref{['eq:toda']} for several polynomial degrees $k$ and several values of $s_Q$ and $s_{\Pi}$.
• Figure 2: Nodal superconvergence in $\tau$ and relative error in the energy balance for the Toda lattice \ref{['eq:toda']} for several polynomial degrees $k$ and several values of $s_Q$ and $s_{\Pi}$.
• Figure 3: Convergence \ref{['fig:rigid_body_varying_degree']}, nodal superconvergence \ref{['fig:rigid_body_varying_degree_different_sampling']} in $\tau$ and relative error in the energy balance \ref{['fig:rigid_body_energybalance']} for the rigid spinning body \ref{['eq:rigid-body']} for several polynomial degrees $k$ and $s_Q = s_{\Pi} = k$.
• Figure 4: Convergence \ref{['fig:damped_wave_nu0_irregular_friction_varying_degree']}\ref{['fig:damped_wave_nu1_irregular_friction_varying_degree']} and nodal superconvergence \ref{['fig:damped_wave_nu1_irregular_friction_varying_degree_different_sampling']} in $\tau$ for several polynomial degrees $k$, $s_Q = k$ and $s_\Pi = 2k$ for the space-discrete quasilinear wave equation with fixed mesh size $h=1$ and $\nu\in\{0,1\}$.
• Figure 5: Convergence in $\tau$ for several space-discretization parameters $h$ (\ref{['fig:damped_wave_nu1_irregular_friction_varying_discretization']}) and relative error in the energy balance with Hamiltonian $\mathcal{H}_h$ (\ref{['fig:damped_wave_nu1_irregular_friction_energybalance']}) for the space-discrete quasilinear wave equation for $\nu=1$.

Theorems & Definitions (19)

• Remark 2.2
• Remark 2.3
• Remark 2.5: classical state dependent port-Hamiltonian systems
• Remark 2.6: structure preserving space discretization
• Lemma 2.7: power balance
• proof
• Remark 2.8
• Example 2.9: Toda lattice
• Example 2.10: spinning rigid body
• Example 2.11: quasilinear wave equation