A comparison of numerical schemes on triangular and quadrilateral meshes

Abstract

Finite volume solution procedures for Euler equations have been extensively studied when applied to quadrilateral meshes. Many methods such as mesh adaptation and multi-grid have been developed which make these procedures efficient and highly accurate. Recently, great interest has been focussed on the development and use of unstructured triangular meshes. One of the advantages of these meshes is that they offer great flexibility in the generation of meshes around extremely complex geometries. The other advantage is that in adaptive mesh refinement the mesh always remains an unstructured triangular mesh, and so no modifications of the basic flow algorithm are required. However, a certain amount of doubt exists in the computational fluid dynamics community concerning the performance of triangular mesh solution schemes; in particular a recent paper by Roe [1] proved that the local truncation error is only first order on very irregular meshes. Giles [2] argues, however, that these triangular schemes can still be globally second order accurate. The goal of this study is to address this question of how well a given computational method can perform on a triangular mesh as compared to the more commonly used quadrilateral meshes. In particular, two node-based finite volume schemes will be examined: the node-based quadrilateral cell Jameson scheme [3] which has been modified for triangular meshes by Mavriplis and Jameson [4], and the quadrilateral cell Lax-Wendroff method developed by Ni [5] which has been modified by Lindquist for use on triangular meshes [6]. Care has been taken to keep the triangular and quadrilateral versions of a scheme similar to provide a fair basis for comparison.