Towards stable shock-capturing via radial basis finite differences on unstructured point clouds

We present a shock-capturing scheme for use in mesh-free simulation. Similar in spirit to essentially non-oscillatory (ENO) schemes, the method computes derivatives on arbitrary point clouds using a combination of finite-difference-like stencils. The method generates several sub-stencils using the radial basis function-finite difference (RBF-FD) method and chooses the smoothest such stencil to compute the derivative. With this, we attempt to ensure stable simulation in the presence of discontinuous flow features like shocks with only modest numerical dissipation. The results are compared to existing methods for solving flows with shocks with RBF-FD, including artificial viscosity approaches. We investigate several benchmark problems, including a forward-facing step, to determine the efficacy of this approach for simulating high-Mach flows.