[Paper Review] Structure-Preserving Time Discretization of Port-Hamiltonian Systems via Discrete Gradient Pairs

Philipp Schulze|arXiv (Cornell University)|Nov 1, 2023

Numerical methods for differential equations4 citations

TL;DR

This paper introduces discrete gradient pairs as a novel structure-preserving time discretization method for nonlinear port-Hamiltonian systems with state-dependent mass matrices. By constructing a discrete gradient pair—specifically the midpoint discrete gradient pair for symmetric, positive definite mass matrices—the method ensures an exact power balance on the time-discrete level, preserving key physical properties in reduced-order models.

ABSTRACT

We discuss structure-preserving time discretization for nonlinear port-Hamiltonian systems with state-dependent mass matrix. Such systems occur, for instance, in the context of structure-preserving nonlinear model order reduction for port-Hamiltonian systems and, in this context, structure-preserving time discretization is crucial for preserving some of the properties of the time-continuous reduced-order model. For this purpose, we introduce a new class of time discretization schemes which is based on so-called discrete gradient pairs and leads to an exact power balance on the time-discrete level. Moreover, for the special case of a pointwise symmetric and positive definite mass matrix, we present an explicit construction of a discrete gradient pair. Finally, we illustrate the theoretical findings by means of a numerical example, where the time-continuous system is a nonlinear reduced-order model for an advection-diffusion problem.

Motivation & Objective

• Address the challenge of preserving physical structure—especially power balance—in time-discretized nonlinear port-Hamiltonian systems with state-dependent mass matrices.
• Overcome the limitation of standard discrete gradient methods, which require explicit appearance of the Hamiltonian gradient in system equations, by introducing discrete gradient pairs.
• Provide an explicit construction of a discrete gradient pair for the case of pointwise symmetric and positive definite mass matrices.
• Demonstrate the method's effectiveness on a nonlinear reduced-order model of an advection–diffusion problem, showing exact discrete power balance and second-order convergence.
• Enable structure-preserving model order reduction by ensuring time-discrete properties mirror those of the continuous system.

Proposed method

• Introduce the concept of discrete gradient pairs as a generalization of discrete gradients for systems where the Hamiltonian gradient does not appear explicitly in the dynamics.
• Define a discrete gradient pair as a pair of functions that satisfy a discrete mean value property, enabling exact power balance on the time-discrete level.
• For symmetric, positive definite mass matrices, construct the midpoint discrete gradient pair using the midpoint rule applied to the gradient of the Hamiltonian.
• Derive a time discretization scheme based on the discrete gradient pair that preserves the algebraic structure of the port-Hamiltonian system.
• Ensure the time-discrete system satisfies the exact power balance equation: $ \frac{\mathcal{H}(x^{n+1}) - \mathcal{H}(x^n)}{\Delta t} = -y^n \cdot u^n $, up to nonlinear solve errors.
• Implement the scheme using a Newton-type solver with tight tolerances to minimize nonlinear iteration errors in numerical experiments.

Experimental results

Research questions

• RQ1 Can discrete gradient pairs be used to construct structure-preserving time integrators for nonlinear port-Hamiltonian systems with state-dependent mass matrices?
• RQ2 Does the proposed method ensure exact power balance on the time-discrete level, even when the Hamiltonian is non-quadratic?
• RQ3 Can an explicit discrete gradient pair be constructed for the case of symmetric, positive definite mass matrices?
• RQ4 How does the convergence behavior of the discrete gradient pair method compare to classical schemes like the implicit midpoint rule?
• RQ5 To what extent does the method preserve physical properties in nonlinear reduced-order models of PDEs?

Key findings

• The proposed discrete gradient pair method achieves exact power balance on the time-discrete level, as proven in Theorem 4.3, even for non-quadratic Hamiltonians.
• For symmetric, positive definite mass matrices, the midpoint discrete gradient pair provides an explicit and computable construction, as shown in Theorem 4.2.
• Numerical experiments on a nonlinear reduced-order model of an advection–diffusion problem show that the discrete gradient pair method yields a power balance error several orders of magnitude smaller than the implicit midpoint rule.
• The experimental order of convergence of the midpoint discrete gradient pair method is approximately two, matching that of the implicit midpoint rule, as confirmed by convergence plots in Figure 2.
• The method maintains structure preservation without requiring the Hamiltonian gradient to appear explicitly in the system equations, overcoming a key limitation of prior discrete gradient approaches.
• The code for the numerical experiments is publicly available via DOI 10.5281/zenodo.10059715, supporting reproducibility and further research.