Higher-Order Finite Volume Solution Reconstruction on Highly Anisotropic Meshes

Abstract

Numerical experiments have demonstrated that the least-squares solution reconstruction suffers from poor accuracy and conditioning on highly anisotropic meshes. These issues are more severe for meshes with finite curvature that are often encountered in aerodynamic applications. This paper presents a comprehensive analysis of solution reconstruction procedure for second-and higher-order finite volume schemes on anisotropic triangular meshes. For this purpose, two families of anisotropic meshes are considered. The accuracy and conditioning of the reconstruction procedure is investigated for straight meshes, which are typically used in the wake region of viscous flows, using unweighted and weighted leastsquares techniques. We found that the unweighted least-squares system exhibits better conditioning and produces more accurate derivatives using the choice of stencil previously used for isotropic meshes. Similarly, the issues are addressed for triangular meshes varying anisotropically over a curved surface. Our results illustrates the shortcomings of the choice of stencil for this family of meshes. Hence, a new choice of stencil is proposed along with a local coordinate system consistent with the cells' anisotropy that resolves the conditioning and accuracy problems for anisotropic curved meshes.

Stencil selection algorithms for WENO schemes on unstructured meshes

“…Therefore at least K number of elements are required in the stencil, in order to have at least as many equations as unknowns. In practice, however this is not sufficient it is prevalent in most of the approaches [2,11,36,37,48,50,51] in the k-exact least squares framework tend to use more elements (ranging from M = 1.5K to M = 3K ) and thus equations, than the number of the unknowns. The more equations used, the more robust and computationally expensive the scheme becomes.…”

Higher-order Unstructured Finite Volume Methods for Turbulent Aerodynamic Flows

“…Our previous results verify that the least-squares system becomes well-conditioned and the derivatives are reconstructed reasonably accurate on meshes without curvature even at high aspect ratio provided that the reconstruction is performed along the principal axes. 10 However, the simulation of high Reynolds number turbulent flows requires sufficiently accurate polynomial approximation on high aspect ratio meshes with finite curvature. Mavriplis showed that least-squares reconstruction suffers from poor accuracy for second-order cell-centered discretization on unstructured meshes.…”

“…12 Our numerical experiment showed the same type of behavior for higher-order reconstruction on this class of meshes. 10 Instead, we construct a local curvilinear coordinate system at the reference point of each control volume. For this purpose, we define a mapping from the physical space into a tangential-normal coordinate system for cells with high curvature near the walls:…”

High-order finite-volume methods for hyperbolic conservation laws on mapped multiblock grids

“…In the Taylor polynomial (10), the reason we divide x − x g and y − y g by R g is to improve the condition number of matrix M (g). As in [19], we are scaling the columns of the matrix so that they have a similar order of magnitude, independent of the grid spacing. We find that it is not necessary to scale the rows of the matrix, by methods such as weighting stencil cells by distance, as is done in [24].…”

“…Identifying a suitable collection of cells from the original block and its neighbors is therefore not trivial. In the fully unstructured and "mesh-free" computational-fluid-dynamics literature, one technique for reconstruction is least-squares interpolation [3,4,24,15,27,23,19,8], which does not presume any underlying spatial relationship between the values used in the interpolation. This is the approach we take here.…”