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Operator splitting for coupled linear port-Hamiltonian systems

Jan Lorenz, Tom Zwerschke, Michael Günther, Kevin Schäfers

TL;DR

This work develops operator-splitting strategies for coupled linear port-Hamiltonian systems to achieve dissipativity-preserving, second-order time integration while exploiting fine structure such as scalar coupling and multirate potential. By combining Strang splitting with discrete-gradient-based updates, the authors ensure energy balance at the discrete level and derive efficient update schemes for the subsystems. They demonstrate that scalar coupling can reduce cost relative to a full implicit midpoint step, and that multirate impulse methods yield further efficiency when slow and fast dynamics are present, as shown on coupled mass-spring-damper chains. The results indicate practical gains in numerical efficiency without sacrificing energy-dissipative properties, with future directions toward higher-order, energy-balanced decompositions and GARK-based coupling strategies.

Abstract

Operator splitting methods tailored to coupled linear port-Hamiltonian systems are developed. We present algorithms that are able to exploit scalar coupling, as well as multirate potential of these coupled systems. The obtained algorithms preserve the dissipative structure of the overall system and are convergent of second order. Numerical results for coupled mass-spring-damper chains illustrate the computational efficiency of the splitting methods compared to a straight-forward application of the implicit midpoint rule to the overall system.

Operator splitting for coupled linear port-Hamiltonian systems

TL;DR

This work develops operator-splitting strategies for coupled linear port-Hamiltonian systems to achieve dissipativity-preserving, second-order time integration while exploiting fine structure such as scalar coupling and multirate potential. By combining Strang splitting with discrete-gradient-based updates, the authors ensure energy balance at the discrete level and derive efficient update schemes for the subsystems. They demonstrate that scalar coupling can reduce cost relative to a full implicit midpoint step, and that multirate impulse methods yield further efficiency when slow and fast dynamics are present, as shown on coupled mass-spring-damper chains. The results indicate practical gains in numerical efficiency without sacrificing energy-dissipative properties, with future directions toward higher-order, energy-balanced decompositions and GARK-based coupling strategies.

Abstract

Operator splitting methods tailored to coupled linear port-Hamiltonian systems are developed. We present algorithms that are able to exploit scalar coupling, as well as multirate potential of these coupled systems. The obtained algorithms preserve the dissipative structure of the overall system and are convergent of second order. Numerical results for coupled mass-spring-damper chains illustrate the computational efficiency of the splitting methods compared to a straight-forward application of the implicit midpoint rule to the overall system.

Paper Structure

This paper contains 8 sections, 18 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Operator Splitting for Linear PHS
  3. Coupled Linear PHS with Scalar Coupling
  4. Coupled Linear PHS with Multirate Potential
  5. Numerical Results
  6. The singlerate case
  7. The multirate case
  8. Conclusion and Outlook

Figures (3)

  • Figure 1: Coupled mass-spring-damper chain, consisting of two chains with $n_1$ and $n_2$ masses, respectively.
  • Figure 2: Coupled mass-spring-damper chain. Global error at $t_{\mathrm{end}} = 2$ vs. the total number of operations for the implicit midpoint rule ($\circ$) and the Strang splitting ($\triangle$).
  • Figure 3: Coupled mass-spring-damper chain. Global error at $t_{\mathrm{end}} = 2$ vs. the total number of operations for the implicit midpoint rule ($\circ$) and the impulse method with multirate factor $m=10$ ($\triangle$).

Theorems & Definitions (1)

  • Definition 1