Stable and flux-conserved meshfree formulation to model shocks

Accurate shock modeling requires that two critical issues be addressed: (1) correct representation of the essential shock physics, and (2) control of Gibbs phenomenon oscillation at the discontinuity. In this work a stable (oscillation limiting) and flux-conserved formulation under the reproducing kernel particle method is developed for shock modeling. A smoothed flux divergence is constructed under the framework of stabilized conforming nodal integration, which is locally-enriched with a Riemann solution to satisfy the entropy production constraints. This Riemann-enriched flux divergence is embedded into the reproducing kernel formulation through a velocity correction that also provides oscillation control at the shock. The correction is constrained to the shock region by an automatic shock detection algorithm that is constructed using the intrinsic spectral decomposition feature of the reproducing kernel approximation. Several numerical examples are provided to verify accuracy of the proposed formulation.