Moving-interface conservation jump identity (${\partial }_{t}a+\nabla \cdot b=([b]\cdot n-v_n[a]){\delta }_s$)

For a moving level surface, $n=\nabla S/|\nabla S|$, $v_n=-S_t/|\nabla S|$, and the surface delta distribution is ${\delta }_s=|\nabla S|\delta (S)$. Piecewise classical conservation laws acquire a distributional flux defect $[b]\cdot n-v_n[a]$. Vanishing of this defect is the Rankine-Hugoniot condition.