新方法用核函数自动捕捉激波，无需人工设粘性或检测间断。
SPIKE: Stable Physics-Informed Kernel Evolution Method for Solving Hyperbolic Conservation Laws
- 用正则化核函数演化模拟守恒律，平滑穿越激波区
- 在标量与向量守恒律上均准确捕获激波，满足跳跃条件
- 统一框架自动守恒，无需显式激波检测或人工粘性
我们提出稳定物理信息核函数演化（SPIKE）方法，用于求解无粘双曲守恒律的数值计算。SPIKE 解决了一个根本矛盾：强形式残差最小化如何能捕捉含间断的弱解。该方法采用带正则化参数演化的再生核表示，其中Tikhonov正则化提供激波形成过程中的平滑过渡机制，使系统动力学可穿越激波奇异性。该方法自动保持守恒性，追踪特征线，并在统一框架内精确捕捉满足Rankine-Hugoniot条件的激波，无需显式激波检测或人工粘性。在标量和向量守恒律上的数值验证证明了该方法的有效性。
原文摘要 · Abstract (English)
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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