Splitting Techniques for DAEs with port-Hamiltonian Applications

TL;DR

This work develops two structure-aware operator-splitting strategies for port-Hamiltonian DAEs: a dimension-reducing decomposition that leverages coupled-subsystem structure (including private index-2 variables) to transfer ODE splitting convergence results to DAEs, and a J-R energy-based decomposition that separates energy-conserving and dissipative dynamics, with generalized Cayley transforms ensuring discrete energy conservation. For coupled index-1 DAEs, the dimension-reducing approach preserves the convergence rate of the underlying ODE splitting (e.g., Strang yields second order), while the J-R approach yields energy-dissipation-preserving integrators, provided algebraic constraints are handled via regularization or careful subproblem assignment. The paper also introduces a regularization framework for cases where structural assumptions fail and demonstrates the methods on port-Hamiltonian circuit benchmarks, highlighting practical gains in structure preservation, energy consistency, and computational efficiency. This work advances robust, structure-preserving time integration for complex multi-physics networks and points toward extensions to index-2 DAEs and higher-order splitting schemes with energy-aware Cayley transforms.

Abstract

In the simulation of differential-algebraic equations (DAEs), it is essential to employ numerical schemes that take into account the inherent structure and maintain explicit or hidden algebraic constraints without altering them. This paper focuses on operator-splitting techniques for coupled systems and aims at preserving the structure in the port-Hamiltonian framework. The study explores two decomposition strategies: one considering the underlying coupled subsystem structure and the other addressing energy-associated properties such as conservation and dissipation. We show that for coupled index-$1$ DAEs with and without private index-2 variables, the splitting schemes on top of a dimension-reducing decomposition achieve the same convergence rate as in the case of ordinary differential equations. Additionally, we discuss an energy-associated decomposition for index-1 pH-DAEs and introduce generalized Cayley transforms to uphold energy conservation. The effectiveness of both strategies is evaluated using port-Hamiltonian benchmark examples from electric circuits.

Figures (13)

• Figure 1: Coupled system consisting of two subsystems (S).
• Figure 2: Example \ref{['ex:D_index1']}: LC-oscillator with coupling current $\jmath_{co}$, Bartel_2023aa. Parameters: $C_i= 10^{-5}$ F, $R_i= 10$$\Omega$ and $L_i= 0.2$ H for $i=1,2$; interval $\mathbb{I}=[0,0.2]$ with the initial values $e_1(0)=e_4(0)=0.1$ V, and $\jmath_1(0)=\jmath_2(0)=1$ A for the dynamic variables; and consistent values for the algebraic variables $e_2(0)=e_3(0)=-9.9$ V and $j_{co}=0$ A.
• Figure 3: Example \ref{['ex:D_index1']}. Top: Reference (analytic) solution for node potentials $e_1$, $e_4$ (left) and currents $\jmath_1$, $\jmath_2$ (right). Middle: Convergence behavior of differential (left) and algebraic variables (right) in some splitting schemes (Lie-Trotter, Strang, Triple-Jump) with appropriate flux approximation for relative step size $h$. Bottom: Hamiltonian (left) and the respective $L^2$-error $\|\mathcal{H}(x_{\mathrm{ref}})-\mathcal{H}(x_{\mathrm{apx},h})\|_{L^2(\mathbb{I})}$ for different numerical approximations (right).
• Figure 4: Example \ref{['ex:D_index2']}: Circuit for two (short) transmission lines with controlled $u_1(t,e_{11})=0.5e_{11}\sin(2\cdot 10^{3}t)$ and $u_2(t,e_{21})=0.5e_{21}\sin(10^{3}t)$ as well as independent voltage sources $v_1(t)=\sin(10^3 t)$ V and $v_2(t)=\sin(3\cdot 10^{3}t)$ V, $t\in \mathbb{I}=[0,4\cdot 10^{-3}]$ and zero initial values except for $e_{21}(0)=-1$ V and $(\jmath,\jmath_{v1},\jmath_{v2})(0)=(1,\tfrac{1}{5},\tfrac{6}{5})$ A. Parameters: $C= 10^{-5}$ F, $R_c= 1$$\Omega$ and $L= 10^{-2}$ H for the coupling part, moreover $C_1= 2\cdot 10^{-4}$ F, $C_2= 4\cdot 10^{-4}$ F, $R_1= 10^{2}$$\Omega$ and $R_2= 5\cdot 10^{1}$$\Omega$ for the private substructures.
• Figure 5: Example \ref{['ex:D_index2']}. Top: Reference for node potential $e_{21}$ (index-1 coupling variable) and convergence behavior of coupling variables. Middle: Reference for substructure variables of subsystem 2: node potentials (left), currents (right). Bottom: Convergence behavior of substructure variables for subsystem 1 (left), subsystem 2 (right). Approaches based on Strang splitting: Radau-IIA method (s=2) with subproblem $i$ (Rad2A S$i$) in last splitting step; midpoint rule for coupling part (3mid), on top Radau-IIA for substructures (3mid+Ra).

Theorems & Definitions (32)

• Definition 2.1: pH-DAE
• Proposition 2.2: beattie2018schaft2018, e.g.
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Proposition 3.1
• Corollary 3.2
• Remark 3.3
• Remark 3.4
• Proposition 3.5