Energy-preserving iteration schemes for Gauss collocation integrators
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909579086921728 |
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| author | Maier, Stefan Marheineke, Nicole Frommer, Andreas |
| author_facet | Maier, Stefan Marheineke, Nicole Frommer, Andreas |
| contents | In this work, we develop energy-preserving iterative schemes for the (non-)linear systems arising in the Gauss integration of Poisson systems with quadratic Hamiltonian. Exploiting the relation between Gauss collocation integrators and diagonal Padé approximations, we establish a Krylov-subspace iteration scheme based on a $Q$-Arnoldi process for linear systems that provides energy conservation not only at convergence --as standard iteration schemes do--, but also at the level of the individual iterates. It is competitive with GMRES in terms of accuracy and cost for a single iteration step and hence offers significant efficiency gains, when it comes to time integration of high-dimensional Poisson systems within given error tolerances. On top of the linear results, we consider non-linear Poisson systems and design non-linear solvers for the implicit midpoint rule (Gauss integrator of second order), using the fact that the associated Padé approximation is a Cayley transformation. For the non-linear systems arising at each time step, we propose fixed-point and Newton-type iteration schemes that inherit the convergence order with comparable cost from their classical versions, but have energy-preserving iterates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10211 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Energy-preserving iteration schemes for Gauss collocation integrators Maier, Stefan Marheineke, Nicole Frommer, Andreas Numerical Analysis 37Mxx, 47J25, 65Lxx, 65Pxx In this work, we develop energy-preserving iterative schemes for the (non-)linear systems arising in the Gauss integration of Poisson systems with quadratic Hamiltonian. Exploiting the relation between Gauss collocation integrators and diagonal Padé approximations, we establish a Krylov-subspace iteration scheme based on a $Q$-Arnoldi process for linear systems that provides energy conservation not only at convergence --as standard iteration schemes do--, but also at the level of the individual iterates. It is competitive with GMRES in terms of accuracy and cost for a single iteration step and hence offers significant efficiency gains, when it comes to time integration of high-dimensional Poisson systems within given error tolerances. On top of the linear results, we consider non-linear Poisson systems and design non-linear solvers for the implicit midpoint rule (Gauss integrator of second order), using the fact that the associated Padé approximation is a Cayley transformation. For the non-linear systems arising at each time step, we propose fixed-point and Newton-type iteration schemes that inherit the convergence order with comparable cost from their classical versions, but have energy-preserving iterates. |
| title | Energy-preserving iteration schemes for Gauss collocation integrators |
| topic | Numerical Analysis 37Mxx, 47J25, 65Lxx, 65Pxx |
| url | https://arxiv.org/abs/2504.10211 |