Operator Splitting Methods for Numerical Solutions of Ordinary Differential Equations

TL;DR

The paper develops operator-splitting methods for approximating Koopman generators associated with nonlinear flows, leveraging Lie-Trotter, Strang, and higher-order exponential splittings within strongly and bi-continuous semigroup frameworks. It employs bi-continuous Chernoff product formulas to guarantee well-posedness and contraction, and introduces coordinate-free, freezing-based algorithms that reduce subproblems to one-dimensional solves. A systematic analysis of exponential-term counts in higher dimensions, along with explicit algorithms for 2D and 3D problems, enables practical high-order schemes. Numerical experiments on Lotka-Volterra, Van der Pol, and Lorenz systems demonstrate favorable accuracy-effort trade-offs and the preservation of key qualitative behavior, illustrating the method’s potential for efficient, structure-preserving ODE integration.

Abstract

We study operator-splitting schemes for approximating Koopman generators of linear semigroups induced by nonlinear flows, a framework originating with Dorroh and Neuberger. Building on ideas of Lie, Kowalewski, and Gröbner, we analyze the Koopman semigroup generated by the Lie-Koopman operator and exploit decompositions of this operator into finitely many components to construct Lie-Trotter, Strang, and higher-order compositions with explicit error bounds. A bi-continuous Chernoff extension guarantees well-posedness and contraction of the splitting operators. Numerical experiments on Lotka-Volterra, Van der Pol, and Lorenz systems validate the theory and demonstrate efficiency via work-precision comparisons. The algorithms remain conceptually simple, relying on coordinate freezing combined with one-dimensional solves, which reflects the classical separation-of-variables principle.

Figures (6)

• Figure 1: Lie-Trotter (Method) and Runge-Kutta (RK45) solutions ($t=100, n=1000$).
• Figure 2: Work-precision comparison for operator-splitting schemes applied to the Lotka-Volterra system. The RMSE relative to a high-accuracy RK45 reference solution is plotted against runtime (seconds) for splitting methods of different orders. Each point corresponds to a fixed number of uniform time steps $n$.
• Figure 3: Phase-space trajectories of the Van der Pol oscillator ($t=25, n=125$) computed using an operator-splitting scheme (dashed red) and a high-accuracy RK45 reference solution (solid blue). The close agreement demonstrates that the splittingmethod accurately reproduces the limit-cycle dynamics.
• Figure 4: Comparison of RMSE vs runtime (seconds) for various operator splitting schemes applied to the Van der Pol Oscillator. Each point represents a step size $n$, showing the trade off between accuracy (RMSE) and runtime.
• Figure 5: Phase-space trajectory of the Lorenz system computed using the Lie-Trotter splitting scheme (blue) and a high-accuracy RK45 reference solution (red) for $t=20$ with $n=20000$ uniform time steps.

Theorems & Definitions (17)

• Example 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.4
• Theorem 4.1: Chernoff's bi-continuous product formula JD3
• Definition 4.2