Nonlinear Port-Hamiltonian Systems: Energy-Consistent Petrov-Galerkin Time Discretization

Core Concepts

This work develops a high-order energy-consistent continuous Petrov-Galerkin (cPG) time discretization scheme for a general class of nonlinear port-Hamiltonian systems, which preserves the energy behavior of the continuous-time solutions.

Abstract

The authors consider a general class of nonlinear port-Hamiltonian systems, which can be formulated as evolution equations with a Hamiltonian function representing the energy in the system. The key contributions are: The authors introduce a continuous Petrov-Galerkin (cPG) time discretization scheme of arbitrary polynomial degree that is energy-consistent, meaning the Hamiltonian of the approximate solutions behaves in the same way as the Hamiltonian of the continuous-time solutions. The energy consistency is achieved by a specific design of the cPG method, which involves the use of an L2-projection mapping. This allows the scheme to be energy-consistent for general Hamiltonian functions, not just quadratic ones. The framework covers a wide range of nonlinear port-Hamiltonian systems, including Hamiltonian systems, gradient systems, and examples such as the quasilinear wave equation, doubly nonlinear parabolic equations, and the Allen-Cahn equation. Numerical experiments are presented to verify the energy consistency and investigate the convergence behavior of the proposed scheme. The main novelty is the development of a high-order energy-consistent cPG method that is applicable to a general class of nonlinear port-Hamiltonian systems, without requiring convexity or quadratic structure of the Hamiltonian.

Stats

None.

Quotes

None.

Key Insights Distilled From

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

by Jan Giesselm... at arxiv.org 04-22-2024

https://arxiv.org/pdf/2404.12480.pdf

Deeper Inquiries

How can the proposed energy-consistent cPG scheme be extended to handle port-Hamiltonian systems with state-dependent operator C(z)

To extend the proposed energy-consistent continuous Petrov-Galerkin (cPG) scheme to handle port-Hamiltonian systems with state-dependent operator C(z), we need to consider the additional complexity introduced by the dependence of the operator on the state variable. In this case, the operator C(z) acts on the time derivative ∂tz, leading to a more intricate formulation of the system dynamics. One approach to address this extension is to reformulate the system equations to incorporate the state-dependent operator C(z) explicitly. By introducing suitable assumptions and conditions on the operator C(z) and its interaction with the other components of the system, such as the Hamiltonian function and the dissipative and control terms, we can modify the cPG scheme to accommodate this additional complexity. Specifically, we would need to adapt the energy-consistency criteria to ensure that the Hamiltonian of the approximate solutions remains consistent with the energy behavior of the system, considering the influence of the state-dependent operator C(z). This may involve adjusting the numerical discretization scheme, the choice of basis functions, and the treatment of the nonlinear terms to account for the state dependence of the operator. In summary, extending the energy-consistent cPG scheme to handle port-Hamiltonian systems with state-dependent operator C(z) requires a careful analysis of the system dynamics, the operator characteristics, and the numerical discretization approach to maintain energy consistency while incorporating the additional complexity introduced by the state dependence of the operator.

What are the implications of the Ge-Marsden theorem on the design of energy-consistent schemes, and how does the proposed approach relate to this result

The Ge-Marsden theorem has significant implications for the design of energy-consistent schemes, particularly in the context of Hamiltonian systems. The theorem states that energy-consistent schemes with fixed time step sizes cannot be symplectic for Hamiltonian systems. This result highlights a fundamental limitation in achieving both energy conservation and symplecticity simultaneously in numerical schemes for Hamiltonian systems. In the context of the proposed energy-consistent cPG scheme for port-Hamiltonian systems, the implications of the Ge-Marsden theorem suggest that while the scheme prioritizes energy consistency, it may not necessarily preserve symplecticity. By focusing on energy preservation and structure preservation properties related to the Hamiltonian function, the cPG scheme aligns with the goal of maintaining energy behavior in the numerical solutions. The proposed approach relates to the Ge-Marsden theorem by acknowledging the trade-off between energy consistency and symplecticity in numerical schemes for Hamiltonian systems. By prioritizing energy conservation and structure preservation, the cPG scheme offers a tailored solution that addresses the challenges posed by the Ge-Marsden theorem and provides a framework for energy-consistent time discretization of port-Hamiltonian systems.

Can the energy-consistent cPG framework be adapted to handle port-Hamiltonian systems with irregular solutions, such as those exhibiting shock behavior

Adapting the energy-consistent cPG framework to handle port-Hamiltonian systems with irregular solutions, such as those exhibiting shock behavior, presents several challenges and considerations. Irregular solutions introduce discontinuities or rapid changes in the system dynamics, which can impact the energy behavior and numerical stability of the scheme. To address irregular solutions, such as shocks, in the context of the cPG framework, specialized numerical techniques may be required. This could involve incorporating shock-capturing methods, adaptive mesh refinement strategies, or high-order numerical schemes to accurately capture the discontinuities and ensure energy consistency in the presence of irregular behavior. Furthermore, the treatment of irregular solutions in the cPG framework may involve modifying the basis functions, refining the time discretization strategy, and enhancing the numerical stability of the scheme to handle the abrupt changes in the system dynamics. Robust error estimation techniques and sensitivity analysis may also be employed to assess the impact of irregular solutions on the energy behavior of the numerical solutions. In summary, adapting the energy-consistent cPG framework to handle port-Hamiltonian systems with irregular solutions requires a tailored approach that accounts for the unique characteristics of irregular behavior, such as shocks. By incorporating specialized numerical techniques and ensuring robustness in the numerical scheme, the framework can be extended to address irregular solutions while maintaining energy consistency in the system dynamics.