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

TL;DR

SPIKE is a novel physics-informed kernel method that accurately solves hyperbolic conservation laws with shocks, using regularization to handle discontinuities without explicit shock detection.

Contribution

The paper introduces SPIKE, a new kernel-based approach that effectively captures shocks in hyperbolic laws through regularized residual minimization, avoiding artificial viscosity.

Findings

Successfully captures shocks satisfying Rankine-Hugoniot conditions

Maintains conservation and tracks characteristics automatically

Validated on scalar and vector conservation laws

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.