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

Abstract

This paper introduces the use of Moving Least-Squares (MLS) approximations for the development of high order upwind schemes on unstructured grids, applied to the numerical solution of the compressible Navier-Stokes equations. This meshfree interpolation technique is designed to reproduce arbitrary functions and their succesive derivatives from scattered, pointwise data, which is precisely the case of unstructured-grid finite volume discretizations. The Navier-Stokes solver presented in this study follows the ideas of the generalized Godunov scheme, using Roe's approximate Riemann solver for the inviscid fluxes. Linear, quadratic and cubic polynomial reconstructions are developed using MLS to compute high order derivatives of the field variables. The diffusive fluxes are computed using MLS as a global reconstruction procedure. Various examples of inviscid and viscous flow are presented and discussed.

On the accuracy of finite volume and discontinuous Galerkin discretizations for compressible flow on unstructured grids

“…For acoustic applications, Eldredge, Colonius and Leonard have used a vortex particle method for the calculation of a corotating vortex pair [6]. In [7][8][9] a method that uses a meshfree technique (moving least squares (MLS)) has been proposed to calculate high-order derivatives of the field variables, achieving a truly multidimensional high-order approach with a FV scheme. In this method, the spatial FV discretization uses the MLS approximation as a kind of 'shape functions' for unstructured grids.…”

A-posteriori error estimation for the finite point method with applications to compressible flow

“…While these techniques can be implemented in a relatively simple manner in structured grids, they cannot be easily extended to unstructured grids and general discretization methods. Applications of p-extrapolation can be found in [33][34][35].…”

“…Note that ϕ * could also be obtained by using higher-order schemes [33][34][35]. Since this approach is attractive in a meshless context, a preliminary investigation was conducted.…”

Assessment of high-order finite volume methods on unstructured meshes for RANS solutions of aeronautical configurations

“…For the evaluation of the viscous fluxes the extrapolated interface variables U ± i,a and their unlimited gradients ∇U ± i,a from the k-exact least square reconstruction of equation (12)a r e averaged from two discontinuous states as detailed in [3,47]. Although, other methods for the evaluation of the diffusive fluxes such as the one of the generalised Riemann problem of Gassner et al [48], or the diffusive flux of direct DG [49,50]c a nb ea p p l i e d ,t h ek -o r d e ra c c u r a t efl u xi si m p l e m e n t e da si te a s i l yo b t a i n e d through element centred reconstructions and has been applied for various flows problems [47,51,52].…”