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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2511.02131 |
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| _version_ | 1866911247984754688 |
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| author | Tapley, Benjamin Kwanen |
| author_facet | Tapley, Benjamin Kwanen |
| contents | We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method exploits homogeneous symmetries to evaluate the projection exactly and in closed form. This yields explicit invariant-preserving integrators for a broad class of nonlinear systems, as well as pseudo-invariant-preserving schemes capable of preserving multiple invariants to arbitrarily high precision. The resulting methods are high-order and introduce negligible computational overhead relative to the base solver. When incorporated into adaptive solvers such as Dormand-Prince 8(5,3), they provide error-controlled, invariant-preserving, high-order time-stepping schemes. Numerical experiments on double-pendulum and Kepler ODEs as well as semidiscretised KdV and Camassa-Holm PDEs demonstrate substantial improvements in both accuracy and efficiency over standard approaches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02131 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit invariant-preserving integration of differential equations using homogeneous projection Tapley, Benjamin Kwanen Numerical Analysis Computational Physics 65L05 (Primary), 65M12 (Secondary) G.1.8 We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method exploits homogeneous symmetries to evaluate the projection exactly and in closed form. This yields explicit invariant-preserving integrators for a broad class of nonlinear systems, as well as pseudo-invariant-preserving schemes capable of preserving multiple invariants to arbitrarily high precision. The resulting methods are high-order and introduce negligible computational overhead relative to the base solver. When incorporated into adaptive solvers such as Dormand-Prince 8(5,3), they provide error-controlled, invariant-preserving, high-order time-stepping schemes. Numerical experiments on double-pendulum and Kepler ODEs as well as semidiscretised KdV and Camassa-Holm PDEs demonstrate substantial improvements in both accuracy and efficiency over standard approaches. |
| title | Explicit invariant-preserving integration of differential equations using homogeneous projection |
| topic | Numerical Analysis Computational Physics 65L05 (Primary), 65M12 (Secondary) G.1.8 |
| url | https://arxiv.org/abs/2511.02131 |