Order of accuracy study of unstructured grid finite volume upwind schemes

Abstract

A detailed numerical study is presented of the order of accuracy of some proposed cell centered, finite volume schemes used for the solution of the 2-D gasdynamic equations on triangular unstructured grids. The schemes studied are based on a MUSCL-type linear reconstruction of interface properties, which seeks to achieve 2nd-order accuracy in space. They are also nominally flux-vector splitting-type schemes, and the results here presented use Liou’s AUSM+ algorithm. The basic aspects effecting the scheme’s order of accuracy are the form in which the reconstruction process is designed and the form in which the limiting process is performed. Two basic concepts are tested with regard to the reconstruction process, namely the use of 1-D-type and gradient-based reconstruction. The limiter can also be constructed as a 1-D-type limiter or as a truly multi-dimensional limiter. The schemes are tested on a linear convection-like model equation and the numerical solutions are compared to the analytical solution, for different mesh sizes, in order to assess the scheme’s order of accuracy. For comparison purposes, the results obtained with a centered scheme are also presented. Second-order accuracy is shown to be only obtained for the centered scheme. The nominally 2nd-order upwind algorithms lead to actual orders of accuracy, which vary from 0.9 to 1.5.