A structure-preserving parametric approximation for anisotropic geometric flows via an $α$-surface energy matrix
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911597005373440 |
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| author | Bao, Weizhu Li, Yifei Ying, Wenjun Zhang, Yulin |
| author_facet | Bao, Weizhu Li, Yifei Ying, Wenjun Zhang, Yulin |
| contents | We propose a structure-preserving parametric approximation for geometric flows with general anisotropic effects. By introducing a hyperparameter $α$, we construct a unified surface energy matrix $\hat{\boldsymbol{G}}_k^α(θ)$ that encompasses all existing formulations of surface energy matrices, and apply it to anisotropic curvature flow. We prove that $α=-1$ is the unique choice achieving optimal energy stability under the necessary and sufficient condition $3\hatγ(θ)\geq\hatγ(θ-π)$, while all other $α\neq-1$ require strictly stronger conditions. The framework extends naturally to general anisotropic geometric flows through a unified velocity discretization that ensures energy stability. Numerical experiments validate the theoretical optimality of $α=-1$ and demonstrate the effectiveness and robustness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24875 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A structure-preserving parametric approximation for anisotropic geometric flows via an $α$-surface energy matrix Bao, Weizhu Li, Yifei Ying, Wenjun Zhang, Yulin Numerical Analysis We propose a structure-preserving parametric approximation for geometric flows with general anisotropic effects. By introducing a hyperparameter $α$, we construct a unified surface energy matrix $\hat{\boldsymbol{G}}_k^α(θ)$ that encompasses all existing formulations of surface energy matrices, and apply it to anisotropic curvature flow. We prove that $α=-1$ is the unique choice achieving optimal energy stability under the necessary and sufficient condition $3\hatγ(θ)\geq\hatγ(θ-π)$, while all other $α\neq-1$ require strictly stronger conditions. The framework extends naturally to general anisotropic geometric flows through a unified velocity discretization that ensures energy stability. Numerical experiments validate the theoretical optimality of $α=-1$ and demonstrate the effectiveness and robustness. |
| title | A structure-preserving parametric approximation for anisotropic geometric flows via an $α$-surface energy matrix |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2512.24875 |