B-stability of numerical integrators on Riemannian manifolds
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909098398711808 |
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| author | Arnold, Martin Celledoni, Elena Çokaj, Ergys Owren, Brynjulf Tumiotto, Denise |
| author_facet | Arnold, Martin Celledoni, Elena Çokaj, Ergys Owren, Brynjulf Tumiotto, Denise |
| contents | We propose a generalization of nonlinear stability of numerical one-step integrators to Riemannian manifolds in the spirit of Butcher's notion of B-stability. Taking inspiration from Simpson-Porco and Bullo, we introduce non-expansive systems on such manifolds and define B-stability of integrators. In this first exposition, we provide concrete results for a geodesic version of the Implicit Euler (GIE) scheme. We prove that the GIE method is B-stable on Riemannian manifolds with non-positive sectional curvature. We show through numerical examples that the GIE method is expansive when applied to a certain non-expansive vector field on the 2-sphere, and that the GIE method does not necessarily possess a unique solution for large enough step sizes. Finally, we derive a new improved global error estimate for general Lie group integrators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_08261 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | B-stability of numerical integrators on Riemannian manifolds Arnold, Martin Celledoni, Elena Çokaj, Ergys Owren, Brynjulf Tumiotto, Denise Numerical Analysis Primary: 65L20. Secondary: 34C40, 53-08, 58D17, 65L05 We propose a generalization of nonlinear stability of numerical one-step integrators to Riemannian manifolds in the spirit of Butcher's notion of B-stability. Taking inspiration from Simpson-Porco and Bullo, we introduce non-expansive systems on such manifolds and define B-stability of integrators. In this first exposition, we provide concrete results for a geodesic version of the Implicit Euler (GIE) scheme. We prove that the GIE method is B-stable on Riemannian manifolds with non-positive sectional curvature. We show through numerical examples that the GIE method is expansive when applied to a certain non-expansive vector field on the 2-sphere, and that the GIE method does not necessarily possess a unique solution for large enough step sizes. Finally, we derive a new improved global error estimate for general Lie group integrators. |
| title | B-stability of numerical integrators on Riemannian manifolds |
| topic | Numerical Analysis Primary: 65L20. Secondary: 34C40, 53-08, 58D17, 65L05 |
| url | https://arxiv.org/abs/2308.08261 |