Adaptive edge-based finite element schemes for the Euler and Navier-Stokes equations on unstructured grids

This paper describes recent developments of high resolution finite element schemes for the solution of the unsteady compressible Euler and Navier-Stokes equations on unstructured meshes. These finite element algorithms use an edge-based data structure, as opposed to a more traditional element-based data structure. The advantage of using such an edge-based data structure is that it provides a unified approach in which the relation between centered and upwind schemes becomes apparent, improves the efficiency of the algorithms, and reduces the storage requirements. A variety of numerical schemes using such edgebased data structure, ranging from Godunov schemes to centered schemes with blended dissipation, is presented and discussed. Adaptive mesh refinement is then added to these solvers to enhance the solution accuracy and efficiency. Various numerical results for a wide range of flow conditions, from subsonic to hyperaonic in both 2D and 30, are presented to demonstrate the performance and versatility of the proposed schemes. -' 1. I N T R O D U C T I O N In recent years, significant progress has been made on developing numerical algorithms for the sclution of the compresible Euler and Navier-Stokes equations. The use of unstructured meshes for computational fluid dynamica problems has become widespread due to their ability to discretize arbitrarily complex geometries and the ease with which mesh adaption can he carried out to improve the solution. However, any numerical schemes based on unstructured meshes require a storage of mesh connectivity information. This requirement leads to an increase of computer memory and the use of indirect addressing to retrieve nearest neighbor information, which, in turn, implies that any numerical algorithms will run slower on unstructured grids than on structured grids. To reduce indirect addressing, finite element schemes based on edge-based data structures have been introduced [l-41. In addition, more sophisticated data structures such as stars, super edges, and Copyright 1993 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. e chains have recently been developed (51. The use of edge-based data structure has been shown to yield significant computational savings for three dimensional problems. Extensive research has been performed during the last few years on upwind algorithms for the solution of the Euler and Navier-Stokes equations on unstructured meshes [6-91. A significant advantage of any upwind discretization is that it is naturally dissipative, as compared with central-difference discretizations, and consequently does not require any problemdependent parameters to adjust. So far, all the upwind schemes implemented as either node-centered or cell-centered discretization8 on unstructured meshes use the finite volume approach and the control volume must be constructed first. In terms of computational efficiency, node-centered schemes are preferable to their cell-center counterparts. In the node-centered approach [6,8], the control volume is typically taken to he part of the neighboring cells that have a vertex at that node. In two dimensions, the part of the cells taken is determined by connecting the centroid of the cell and the midpoints of the two edges that share the node. In 2-D, tbe part of the cells taken is determined by a surface that is constructed in a similar way, a somewhat complicated geometrical process in three dimensions. The switching from element to edge-based data structure enables the implementation of upwind schemes trivial and straightforward in the context of finite elements. This is especially attractive for three dimensional problems, as there is no need to construct control volumes explicitly and geometrically. The objective of this paper is to present recently developed high accuracy schemes on unstructured grids using an edge-based data structure. This edgebased data structure provides a unified approach in which the link between centered and upwind schemes becomes apparent. The use of such an edge-based data structure not only improves the efficiency of the algorithms, but also enables a straightforward implementation of upwind schemes in the context of finite element methods. A variety of numerical schemes using the edge-based data structure is presented and the performance of these schemes in terms of solution accuracy and overall computational efficiency is discussed. Some different strategies for the discretization of the viscous terms are considered. An approach well suited for use with an edge-based data structure is then introduced and presented. An E-refinementfcoarsening adaptive scheme is implemented in these schemes to enhance the solution accuracy and efficiency. Various numerical examples for a wide range of flow conditions, from subsonic to hypersonic in both 2D and 3D, are presented to demonstrate the performance and versatility of the proposed algorithms. 2. GOVERNING EQUATIONS The Navis:-Stokes equations governing unsteady compressible viscous flows can be expressed in the conservative form as