ABSTRACT
Method John A. Ekaterinaris, School of Mechanical and Aerospace Engineering, University of Patras, Greece and Foundation for Research and Technology-Hellas, Institute of Applied and Computational Mathematics, 71110 Heraklion, Greece, ekaterin@iacm.forth.gr
Generation of structured three-dimensional meshes even in domains with moderate complexity, such as a wing body junction, is not trivial. On the other hand, the performance of most high-resolution accurate methods depends on the smoothness of the grid. The difficulty in generating smooth structured grids for complex geometries has promoted the development of finite-volume algorithms for unstructured grids [83], [108], [131], [186], [191]. These unstructured grid methods are second order accurate. Higher-order finite-volume schemes were pioneered by Barth and Frederickson [16] with the k-exact finite volume scheme that can be used for arbitrary high order reconstruction in triangular or tetrahedral meshes. The implementation of ENO for unstructured grids was developed by Abgrall [1], while WENO schemes for triangular meshes were developed by Friedrich [58] and Hu and Shu [90]. Theoretically, the finite volume approaches of [16] and [90], [1] can be used to obtain arbitrarily high-order accurate finite volume schemes with high-order polynomial data reconstructions. However, in practice higher than linear reconstructions are not used in three dimensions because of the difficulty to construct nonsingular stencils and the large memory required to store the reconstruction coefficients. This was clearly shown by Delanaye and Liu [45] who found that for the third-order (quadratic reconstruction) FV scheme in three dimensions, the average size of the reconstruction stencils is about 5070. The size of reconstruction stencils increases nonlinearly with the order of accuracy and it was estimated that for the fourth-order FV scheme the stencil size would be approximately 120.
