Analysis and Improvement of Upwind and Centered Schemes on Quadrilateral and Triangular Meshes

Second-order accurate upwind and centered schemes are presented in a framework that facilitates their analysis and comparison. The upwind scheme employed consists of a reconstruction step (Van Leer 1977) followed by an upwind step (Roe 1981). The two centered schemes are of Lax-Friedrichs (L-F) type. They are the nonstaggered versions of the N-T scheme (called ORD in Nessyahu-Tadmor 1990) and the CE/SE method with = 1=2 (Chang 1995). The upwind scheme is extended to the case of two spatial dimensions (2D) in a straightforward manner. The N-T and CE/SE schemes are extended in a manner similar to the 2D extensions of the CE/SE schemes by Wang and Chang (1999) and Zhang et al. (2002); the slope estimates, however, are simpli ed. Fourier stability and accuracy analyses are carried out for these schemes for the standard 1D and the 2D quadrilateral mesh cases. In the nonstandard case of a triangular mesh, the triangles must be paired up when analyzing the upwind and N-T schemes. An observation resulting in an extended N-T scheme which is faster and uses only one third of the storage for ow data compared with the CE/SE method is presented. Numerical results are shown. Other improvements to the schemes are discussed. Introduction. When solving a uid ow problem, a researcher has the option of choosing between upwind and centered schemes using a quadrilateral or a triangular mesh. In this paper, trade-o s among these choices are discussed. Relations between schemes and their strengths and weaknesses are shown, and improvements are suggested. The schemes employed are among the simplest second-order accurate schemes that can capture shocks and deal with unsteady problems. Both the upwind and centered schemes here use piecewiselinear reconstructions, i.e., MUSCL interpolants (monotone upwind schemes for conservation laws, Van Leer, 1977), which extend Godunov's piecewise constant method (1959). The key di erence is that for the upwind scheme, numerical dissipation is added by the upwind step (Roe 1981, 1986), while for the centered schemes, dissipation is obtained by, loosely put, averaging the neighboring data (scheme ORD of Nessyahu and Tadmor 1990). The upwind step has a few drawbacks. Roe's ux-di erence splitting, which is mathematically rigorous and among the most popular, may cause oscillations as in the case of a slow-moving shock, or instability as in the carbuncle problem. The AUSM scheme (Liou and Ste en Jr. 1992, Wada and Liou 1997) does not have these problems, but it is not clear to this author why the scheme works. The upwind step is also sometimes perceived to be costly and di cult to grasp. In spite of these problems, upwind schemes are popular because they work well for a large class of ows. The upwind step employed here is Roe's splitting with an entropy correction described in Huynh (1995a). It can be derived by diagonalization and coded by stepping across one acoustic wave. The resulting scheme is concise and economical; the presentation below is also simpler than most presentations in the literature. Numerical solutions obtained with this upwind scheme for the 1D Euler equations can be found in Huynh (1995a,b). The 2D extension of this scheme is conceptually straightforward. For other versions of upwind schemes, see, e.g., Barth and Jespersen (1989), Roe (1989), Hirsch (1990), and Venkatakrishnan (1995). To avoid upwinding, a second-order accurate scheme, which extends the rst-order scheme of Lax-Friedrichs (L-F), was introduced by Nessyahu and Tadmor (1990). There, the reconstruction step is the same as that of the upwind scheme, but 1 16th AIAA Computational Fluid Dynamics Conference 23-26 June 2003, Orlando, Florida AIAA 2003-3541 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. the upwind step is avoided by the use of a staggered mesh. The scheme employed here, however, is the nonstaggered version obtained by overlaying two staggered meshes to form a regular mesh. The drawback of the nonstaggered version is that the computing time doubles. The main reason this version is chosen is that the 2D extension retains its key advantage of simplicity especially when the mesh is triangular. In addition, the mesh and the boundary conditions can be chosen to be the same as those of the upwind scheme. The staggered version, on the other hand, requires two sets of meshes and two sets of boundary conditions|compromising the advantage of simplicity. Moreover, for a triangular mesh, the extension of the staggered version is quite involved (Arminjon et al. 1997) and, as will be discussed, the advantage in computing time may no longer hold. Therefore, unless otherwise stated, we deal only with the the nonstaggered version below. Note that since there is no one-sided bias, the N-T scheme is centered. Also note that these L-F type centered schemes are di erent from the semidiscrete centered schemes made popular by Jameson et al. (1984). While the numerical dissipation of the N-T scheme is a lot less than that of the L-F method, it is still considerable. The CE/SE or conservation element and solution element method (Chang 1995) provides a way to adjust dissipation for these centered schemes. Compared with the N-T scheme, the mesh, the balancing of uxes, and the updates of the cell average quantities are essentially identical. The di erence is in the calculation of the slopes (of the linear interpolant). For CE/SE, the slopes must be stored, and due to the way the slopes are updated, numerical dissipation can be adjusted via a parameter called . For the general CE/SE scheme ( 6= 1=2), the slope calculation is quite di erent from that in a typical MUSCL approach. When = 0, the scheme has no numerical dissipation, i.e., it is reversible in time. Currently, the CE/SE member employed in essentially all practical calculations corresponds to = 1=2. For this reason, we restrict our attention to this member and, from this point on, unless otherwise stated, the term CE/SE refers to the member with = 1=2. Note that there are numerous di erences in terminology between (Nessyahu and Tadmor 1990) and (Chang 1995); here, the terminology in the former is often employed. The CE/SE schemes were extended to 2D for unstructured triangular meshes by Wang and Chang (1999) using the nonstaggered version. What is novel about this extension is that the spatial domain where each reconstruction is valid at the beginning of the time level is a hexagon, which is roughly twice as big as the triangular cell. This CE/SE approach to extension is also applied here to the N-T scheme. (Such an extension was mentioned as a coupled version for the CE/SE scheme in Chang et al. (1999) and has recently been incorporated|independently from this work|as an option in the CE/SE code; private communication with Drs. Ananda Himansu, Ching Y. Loh, and Xiao-Yen Wang. Note, however, that the extended N-T scheme presented here has numerous di erences resulting in a scheme which is faster and requires considerably less storage.) The quadrilateral-mesh extension for the CE/SE method can be found in Zhang et al. (2002); see also Cook (1999). The CE/SE schemes have been applied to solve numerous practical problems in two and three dimensions with a lot of success, especially in aeroacoustics. For a structured quadrilateral mesh, in a manner similar to the 1D case, the nonstaggered mesh in Zhang et al. (2002) can be obtained by overlaying two staggered meshes in Arminjon et al. (1995) and Jiang and Tadmor (1998) (see also Jiang et al. 1998). For a triangular mesh, however, a similar statement does not hold; in fact, the nonstaggered extension in Wang and Chang (1999) appears to have numerous advantages over a staggered-mesh extension (remark (c) in x6 below). In this paper, the schemes involved are rst presented for the 1D advection equation where key ideas and trade-o s can already be seen. It is shown that the N-T and the CE/SE ( = 1=2) schemes are respectively the centered counterparts of Van Leer's rst and second upwind schemes. Next, the extensions of the upwind, N-T, and CE/SE schemes to the 1D Euler equations are explained. Then, extensions to the 2D Euler equations on a quadrilateral and a triangular mesh as well as the simpli ed slope estimates are described. Comparison of schemes via Fourier stability and accuracy analyses are carried out. Here, for a triangular mesh, we must pair up the downward and upward pointing triangles when analyzing the upwind and N-T schemes. Concerning the two L-F type methods, the N-T and CE/SE schemes are shown to produce essentially the same numerical solutions. The former has the advantage of better coupling; consequently, it converges better for a steady state problem. In addition, the following observation yields an extended N-T scheme which is faster and requires consider-