Operator Splitting Methods for Numerical Solutions of Ordinary Differential Equations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915677308190720 |
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| author | Banjara, A. AlJabea, I. Papamarkou, T. Neubrander, F. |
| author_facet | Banjara, A. AlJabea, I. Papamarkou, T. Neubrander, F. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17524 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Operator Splitting Methods for Numerical Solutions of Ordinary Differential Equations Banjara, A. AlJabea, I. Papamarkou, T. Neubrander, F. Numerical Analysis Dynamical Systems 65L05, 65L20, 47D03, 37M20 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. |
| title | Operator Splitting Methods for Numerical Solutions of Ordinary Differential Equations |
| topic | Numerical Analysis Dynamical Systems 65L05, 65L20, 47D03, 37M20 |
| url | https://arxiv.org/abs/2506.17524 |