Meshfree MUSCL schemes with a posteriori order detection: exactness, conservation and admissibility

We collect the mathematical framework of a class of meshfree MUSCL schemes for hyperbolic systems of conservation laws on scattered point clouds, whose implementation and numerical study are reported in a companion paper [11]. Midpoint Riemann problems between moving least-squares (MLS) reconstructions are differentiated by an MLS derivative operator with zero row sums. We prove exactness for polynomial solutions of linear problems, consistency of the design order on Cartesian lattices for an odd derivative degree selected by a parity argument, and an exact balance law that identifies the conservation error with the lack of antisymmetry of the volume-weighted derivative weights. Orienting the Rusanov dissipation by the signs of the derivative weights (the aligned orientation) makes the first-order scheme positivity-preserving for the Euler equations, for derivative weights of any degree. For the per-particle MOOD procedure with three interface rules we characterize the rules under which a particle at the lowest order is advanced by the first-order scheme. Finally we show that stage-wise MOOD inside an arbitrary explicit strong-stability-preserving Runge–Kutta method yields admissible stages and updates for every distribution of the orders, provided that the detection checks the forward Euler substeps of the canonical Shu–Osher form rather than the stage values; for the SSP methods of Heun (RK2) and Shu–Osher (RK3), whose stages contain only U n and the last substep, checking the stage values gives the same guarantee, for SSPRK(5,4) it does not.