High-order meshfree MUSCL schemes with a posteriori order detection for hyperbolic systems on scattered point clouds

We present meshfree MUSCL schemes of arbitrary design order $k$ for hyperbolic systems of conservation laws on scattered point clouds. Moving least-squares (MLS) reconstructions of degree $k-1$ are evaluated at the midpoints between neighbouring particles, the jumps are resolved by approximate Riemann solvers, and the interface fluxes are differentiated by an MLS derivative operator whose weights sum to zero. The scheme is exact for polynomial solutions of linear problems, reaches order $k$ on symmetric stencils, and its conservation error is the lack of antisymmetry of the derivative weights. Orienting the dissipation of the Rusanov flux by the signs of the derivative weights keeps density and pressure positive in the first-order fallback scheme for derivative operators of any degree. The degree of the derivative operator can therefore stay fixed while MOOD lowers only the degree of the reconstructions. Discontinuities are treated by slope limiters or by a per-particle MOOD procedure within SSP Runge--Kutta methods. Numerical experiments confirm the design orders up to five for smooth solutions on regular and perturbed clouds and show that, with a derivative operator of degree one, the conservation defect at moving shocks decreases under refinement. Two-dimensional shock problems up to the double Mach reflection and the Mach 3 forward-facing step are computed robustly with positive density and pressure. A smooth-extremum detection adapted to meshfree stencils prevents MOOD from clipping the core of an isentropic vortex for orders three and four once the core is resolved by ten to twenty particles per radius.