A Conservative Meshless Scheme: General Order Formulation and Application to Euler Equations

In this paper, we present a mesh-free, finite-volume-like scheme designed for numerically solving conservation laws. We first derive a conservative formulation for computing mesh-free first derivatives based on a set of unknown coefficients satisfying polynomial consistency and other conditions. This formulation leads to a generalization that allows flexible choices of flux schemes while remaining locally conservative. We present an algorithm, based on minimum-norm solutions, for calculating unique meshless coefficients given a point distribution and connectivity. Numerical examples of solving the 2D Euler equations demonstrate the flexibility and practicality of the conservative meshless scheme.