SPIKE: Stable Physics-Informed Kernel Evolution Method for Solving Hyperbolic Conservation Laws
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914105102696448 |
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| author | Su, Hua Zhang, Lei Zhao, Jin |
| author_facet | Su, Hua Zhang, Lei Zhao, Jin |
| contents | We introduce the Stable Physics-Informed Kernel Evolution (SPIKE) method for numerical computation of inviscid hyperbolic conservation laws. SPIKE resolves a fundamental paradox: how strong-form residual minimization can capture weak solutions containing discontinuities. SPIKE employs reproducing kernel representations with regularized parameter evolution, where Tikhonov regularization provides a smooth transition mechanism through shock formation, allowing the dynamics to traverse shock singularities. This approach automatically maintains conservation, tracks characteristics, and captures shocks satisfying Rankine-Hugoniot conditions within a unified framework requiring no explicit shock detection or artificial viscosity. Numerical validation across scalar and vector-valued conservation laws confirms the method's effectiveness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18266 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | SPIKE: Stable Physics-Informed Kernel Evolution Method for Solving Hyperbolic Conservation Laws Su, Hua Zhang, Lei Zhao, Jin Numerical Analysis Artificial Intelligence Machine Learning Analysis of PDEs We introduce the Stable Physics-Informed Kernel Evolution (SPIKE) method for numerical computation of inviscid hyperbolic conservation laws. SPIKE resolves a fundamental paradox: how strong-form residual minimization can capture weak solutions containing discontinuities. SPIKE employs reproducing kernel representations with regularized parameter evolution, where Tikhonov regularization provides a smooth transition mechanism through shock formation, allowing the dynamics to traverse shock singularities. This approach automatically maintains conservation, tracks characteristics, and captures shocks satisfying Rankine-Hugoniot conditions within a unified framework requiring no explicit shock detection or artificial viscosity. Numerical validation across scalar and vector-valued conservation laws confirms the method's effectiveness. |
| title | SPIKE: Stable Physics-Informed Kernel Evolution Method for Solving Hyperbolic Conservation Laws |
| topic | Numerical Analysis Artificial Intelligence Machine Learning Analysis of PDEs |
| url | https://arxiv.org/abs/2510.18266 |