SPIKE: Stable Physics-Informed Kernel Evolution Method for Solving Hyperbolic Conservation Laws
This provides a unified framework for computational fluid dynamics and related fields, addressing a fundamental challenge in numerical methods for hyperbolic PDEs.
The paper tackles the problem of solving hyperbolic conservation laws with discontinuities by introducing the SPIKE method, which resolves the paradox of using strong-form residual minimization for weak solutions, achieving stable shock capture without explicit detection or artificial viscosity.
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.