Finite volume solvers and Moving Least-Squares approximations for the compressible Navier–Stokes equations on unstructured grids

This paper explores the approximation power of Moving Least-Squares (MLS) approximations in the context of higher-order finite volume schemes on unstructured grids. The scope of the application of MLS is threefold: (1) computation of high-order derivatives of the field variables for a Godunov-type approach to hyperbolic problems or terms of hyperbolic character, (2) direct reconstruction of the fluxes at cell edges, for elliptic problems or terms of elliptic character, and (3) multiresolution shock detection and selective limiting. A major advantage of the proposed methodology over the most popular existing higher-order methods is related to the viscous discretization. The use of MLS approximations allows the direct reconstruction of high-order viscous fluxes using quite compact stencils, and without introducing new degrees of freedom, which results in a significant reduction in storage and workload. A selective limiting procedure is proposed, based on the multiresolution properties of the MLS approximants, which allows to switch off the limiters in smooth regions of the flow. Accuracy tests show that the proposed method achieves the expected convergence rates. Representative simulations show that the methodology is applicable to problems of engineering interest.