SLERP-TVDRK (STVDRK) Methods for Ordinary Differential Equations on Spheres
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929541052628992 |
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| author | Leung, Shingyu Chau, Wai Ming Lee, Young Kyu |
| author_facet | Leung, Shingyu Chau, Wai Ming Lee, Young Kyu |
| contents | We mimic the conventional explicit Total Variation Diminishing Runge-Kutta (TVDRK) schemes and propose a class of numerical integrators to solve differential equations on a unit sphere. Our approach utilizes the exponential map inherent to the sphere and employs spherical linear interpolation (SLERP). These modified schemes, named SLERP-TVDRK methods or STVDRK, offer improved accuracy compared to typical projective RK methods. Furthermore, they eliminate the need for any projection and provide a straightforward implementation. While we have successfully constructed STVDRK schemes only up to third-order accuracy, we explain the challenges in deriving STVDRK-r for r \ge 4. To showcase the effectiveness of our approach, we will demonstrate its application in solving the eikonal equation on the unit sphere and simulating p-harmonic flows using our proposed method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10420 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | SLERP-TVDRK (STVDRK) Methods for Ordinary Differential Equations on Spheres Leung, Shingyu Chau, Wai Ming Lee, Young Kyu Numerical Analysis We mimic the conventional explicit Total Variation Diminishing Runge-Kutta (TVDRK) schemes and propose a class of numerical integrators to solve differential equations on a unit sphere. Our approach utilizes the exponential map inherent to the sphere and employs spherical linear interpolation (SLERP). These modified schemes, named SLERP-TVDRK methods or STVDRK, offer improved accuracy compared to typical projective RK methods. Furthermore, they eliminate the need for any projection and provide a straightforward implementation. While we have successfully constructed STVDRK schemes only up to third-order accuracy, we explain the challenges in deriving STVDRK-r for r \ge 4. To showcase the effectiveness of our approach, we will demonstrate its application in solving the eikonal equation on the unit sphere and simulating p-harmonic flows using our proposed method. |
| title | SLERP-TVDRK (STVDRK) Methods for Ordinary Differential Equations on Spheres |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2410.10420 |