Accuracy assessment of a high-order moving least squares finite volume method for compressible flows

Abstract

This paper proposes an investigation of some important properties of a high-order finite-volume moving least-squares based method (FV-MLSs) for the solution of two-dimensional Euler and Navier–Stokes equations on unstructured grids. A particular attention is paid to the computation of derivatives of shape functions by means of diffuse or full discretization. Furthermore, we introduce the semi-diffuse approach, which is a compromise in terms of accuracy and computational cost. In addition, we investigate the influence of the curvature of wall boundary conditions on the proposed numerical scheme. As expected, we found an improvement when the walls’ curvature is taken into account. However, and differently as in discontinuous Galerkin schemes, the use of a straight representation of the wall normals does not induce a substantial loss of accuracy. Numerical simulations show the accuracy and the robustness of the numerical approach for both inviscid and viscous flows.

Introduction

Achieving high order of accuracy on unstructured grids remains a great challenge to effectively capture complex flow features in computational fluid dynamics and computational aeroacoustics. During the last decade, high order continuous finite element methods (FEMs), discontinuous Galerkin methods (DGMs) and spectral volume methods (SVMs) have gained popularity for the numerical simulation of compressible Euler and Navier–Stokes equations [40]. On the contrary, the field of research for higher-order unstructured finite-volume schemes is less active, although second-order finite-volume schemes are routinely used for engineering problems. The key issue in the development of high-order unstructured finite-volume schemes is the implementation of efficient reconstruction procedures of unknown variables at the interface of the control volumes.

The most popular approaches employed to construct third- and fourth-order finite-volume scheme for hyperbolic conservation laws on unstructured grids are k-exact least-squares reconstruction [2], [14], [8], [30], [39], [27], [32] and ENO [1], [14], [33] or WENO [12], [17], [36], [9] reconstructions. The former stipulates that any solution which is expressed as a polynomial function of degree k is reconstructed exactly at the cell interface. By its design, k-exact least-squares reconstruction properly ensures conservation of the mean, which requires that the average of the reconstructed solution over the control volume is equal to the average of the original function [32]. The method can be applied to curved geometries without deteriorating the high-order accuracy by means of careful treatment of flux integrations and control volume moments [31]. An efficient implicit matrix-free Newton-GMRES methods for fourth-order k-exact discretization of two dimensional steady Euler equations is developed in [27]. On the other hand, FV-WENO schemes combine the efficiency of their finite-difference counterparts with the flexibility of irregular grids. Third- and fourth-order WENO schemes, based on linear and quadratic polynomials for triangular meshes, are successfully applied to various unsteady inviscid flows in [12], [17]. Dumbser et al. [9] constructed a fourth-order quadrature free WENO scheme for non-linear hyperbolic systems on unstructured tetrahedral meshes for straight geometries.

Recently, Cueto-Felgueroso et al. [6], [7] proposed to evaluate the successive derivatives involved in the high-order reconstruction step by means of moving least squares (MLSs) approximations [22]. The interpolation structure is obtained from a weighted least-squares fitting of the solution based on a given kernel function. As a consequence, the reconstructed solution is not a polynomial. Furthermore, the resulting FV-MLS method allows a direct reconstruction of the high-order viscous fluxes and the mean conservation is automatically satisfied for steady-state computations. A comparative study with a discontinuous Galerkin scheme for the solution of the Navier–Stokes equations on quadrilateral meshes can be found in [29].

This paper proposes an investigation of some important properties of the FV-MLS method for the solution of two-dimensional Euler and Navier–Stokes equations on unstructured triangular grids. First, we focus on the computation of derivatives of the shape function. In previous studies [6], [7], [29], the derivatives of MLS shape functions were approximated using diffuse derivatives [18]. Even though using this approach, the convergence rate of the numerical scheme is conserved, no work has been made about the influence on the accuracy of the method. Here, we investigate the influence of the computation of the derivatives using the diffuse or the full discretization. In addition, we introduce the semi-diffuse approach, which is a compromise in terms of accuracy and computational cost. On the other hand, the influence of curved boundaries on the accuracy of high order methods has been pointed out by many authors [21], [9], [24], [27], [13]. In particular, it has been shown that Discontinuous Galerkin method is particularly sensitive, and straight representation of curved boundaries introduces large errors in the solution [3], [21]. Here, we investigate the influence of the curvature of wall boundary conditions on the formal order of accuracy in the context of moving least squares based finite volume methods. Numerical simulations demonstrate the accuracy and the robustness of the high-order FV-MLS method for both inviscid and viscous flows.

The outline of the paper is as follows: First, the finite-volume formulation based on moving least-squares reconstruction is described in Section 2. Then, the accuracy assessment for smooth inviscid flow around a circular cylinder is shown in Section 3. Finally, numerical experiments for separated viscous flows are presented for various configurations in Sections 4 Numerical applications, 5 Concluding remarks.