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arXiv · 2510.18266

SPIKE: Stable Physics-Informed Kernel Evolution Method for Solving Hyperbolic Conservation Laws

Hua Su·Lei Zhang·Jin Zhao
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Abstract

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.

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BibTeXRIS

Hua Su, Lei Zhang, Jin Zhao. 2025-10-21. SPIKE: Stable Physics-Informed Kernel Evolution Method for Solving Hyperbolic Conservation Laws. https://arxiv.org/abs/2510.18266

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