A Semi-Lagrangian Spherical Essentially Non-Oscillatory (SENO) Scheme for Advection Equations of S2-valued Functions
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866913013653569536 |
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| author | Leung, Shingyu |
| author_facet | Leung, Shingyu |
| contents | We develop a numerical scheme for solving the advection equation of $\mathbb{S}^2$-valued functions of real variables, which models the time-evolution of a $\mathbb{S}^2$-valued mapping on the real line by a known velocity field. The idea is to extend the semi-Lagrangian method for the linear scalar advection equation. We first construct the backward flow map between two adjacent time levels and then interpolate the discrete ordered data of $\mathbb{S}^2$. To handle $\mathbb{S}^2$-functions which have kinks or sharp discontinuity in their components, we incorporate the \textit{Spherical Essentially Non-Oscillatory} (SENO) interpolation method, which effectively reduces the spurious oscillations in high-order reconstructions. We will show multiple examples to demonstrate the accuracy and effectiveness of the proposed algorithm for the partial differential equation of $\mathbb{S}^2$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06587 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Semi-Lagrangian Spherical Essentially Non-Oscillatory (SENO) Scheme for Advection Equations of S2-valued Functions Leung, Shingyu Numerical Analysis We develop a numerical scheme for solving the advection equation of $\mathbb{S}^2$-valued functions of real variables, which models the time-evolution of a $\mathbb{S}^2$-valued mapping on the real line by a known velocity field. The idea is to extend the semi-Lagrangian method for the linear scalar advection equation. We first construct the backward flow map between two adjacent time levels and then interpolate the discrete ordered data of $\mathbb{S}^2$. To handle $\mathbb{S}^2$-functions which have kinks or sharp discontinuity in their components, we incorporate the \textit{Spherical Essentially Non-Oscillatory} (SENO) interpolation method, which effectively reduces the spurious oscillations in high-order reconstructions. We will show multiple examples to demonstrate the accuracy and effectiveness of the proposed algorithm for the partial differential equation of $\mathbb{S}^2$-functions. |
| title | A Semi-Lagrangian Spherical Essentially Non-Oscillatory (SENO) Scheme for Advection Equations of S2-valued Functions |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2604.06587 |