BEHAVIOR OF LINEAR RECONSTRUCTION TECHNIQUES ON UNSTRUCTURED MESHES

This paper presents an assessment of a variety of reconstruction schemes on meshes with both quadrilateral and triangular tessellations, The investigations measure the order of accuracy, absolute error, and convergence properties associated with each method. Linear reconstruction approaches using both Green-Gauss and least-squares gradient estimation are evaluated against a structured MUSCL scheme wherever possible, In addition to examining the influence of polygon degree and reconstruction strategy, results with three limiters are examined and compared against unlimited results when feasible. The methods are applied on quadrilateral, right triangular, and equilateral triangular elements to facilitate an examination of the scheme's behavior on a variety of element shapes. The numerical test cases include feell-noown internal and external inviscid examples and also a supersonic vortex problem for which there exists a closed-form solution to the two-dimensional compressible Euler equations. Such investigations indicate that the least-squares gradient estimation provides significantly more reliable results on poor quality meshes. Furthermore, limiting only the face normal component of the gradient can significantly increase both accuracy and convergence while still preserving the integral cell average and maintaining monoticity. The first-order method performs poorly on stretched triangular meshes, and analysis shows that such meshes result in poorly aligned left and right states for the Riemann problem. The higher average valence of a vertex in the triangular tesselations does not appear to enhance the wave propagation, accuracy, or convergence properties of the method. Typically, quadrilateral elements provide superior or equivalent discrete solutions with approximately 50% fewer edges in the domain two-dimensional. However, on very poor quality meshes, the triangular elements routinely yield superior accuracy as a result of the trapezoidal quadrature of the Galerkin portion of the numerical flux function.