5.2.3 Cell-Centred versus Median-Dual Scheme 单元中心格式与中位对偶格式的比较[cfd-5-2-3]

The relative advantages and disadvantages of the cell-centred and the median-dual scheme are the subject of controversial debates. The main reason is the lack of fair comparisons of the two methodologies with respect to accuracy, computational time and memory for realistic configurations. Our intention here is to collect the most important arguments for and against each of the approaches regarding:

• accuracy,
• computational work,
• memory requirements, and
• flexibility.

This should lead to a greater understanding of the problems inherent to each scheme and should be of help in selecting the most suitable scheme for the intended applications.

A cell-centred scheme on a triangular/tetrahedral grid leads to about twice/six times as many control volumes and hence degrees of freedom as a median-dual scheme [36]. On typical mixed grids, which consist of tetrahedra and prisms, a cell-centred scheme gives roughly three times more unknowns than a median-dual scheme. This suggests that cell-centred schemes are more accurate than cell-vertex discretisations on an identical grid. However, the residual of a cell-centred scheme results from a much smaller number of fluxes as compared to a median-dual scheme (three versus approximately seven on a tetrahedral grid), which may impair the accuracy. Thus, there is no clear evidence about which scheme might be superior.

The median-dual scheme suffers from a particular problem on stretched triangular and tetrahedral grids. Consider, for example, Fig. 5.10, which shows a tessellation composed of right triangles, as it is often employed near solid walls for viscous flows. We can see in Fig. 5.10a that the face \(\Delta S_{ij}\) becomes highly skewed with respect to the edge \(ij\). However, spatial discretisation schemes mostly assume fluxes to be orthogonal to a face (especially Riemann solvers). Thus, an error is introduced which is particularly significant for a first-order scheme [38]. The situation can be improved using the so-called containment-dual control volume [39]. As depicted in Fig. 5.11, the containment-dual approach employs the centres of the minimum spanning circles/spheres instead of the cell centroids to define the faces. This leads to control volumes identical to those on quadrilateral grids (Fig. 5.10b). Notice that there is no face area associated with diagonal edges like \(ij'\). An additional effort is required for pre-processing, but the solution accuracy can be improved noticeably [40]. Of course, another possibility is to employ directly quadrilateral or hexahedral

cells within the boundary layers. Further discussion of grid-induced errors can be found in Ref. [22] and [23].

Another problem inherent to the median-dual scheme is the discretisation at boundaries of the physical domain. What happens is that there is only about one half of the control volume left at the boundary (cf. Fig. 4.6). The integration of fluxes around the faces results in a residual located inside - ideally at the centre - of the control volume. However, the residual is associated with the node, residing directly on the boundary. This mismatch leads to increased discretisation error in comparison to the cell-centred scheme, which is particularly undesirable on solid walls. The definition of the dual control volume causes also problems at sharp corners (like trailing edges), which show up as unphysical peaks in pressure or density. Further complications arise at periodic boundaries (see Chapter 8.8), where the fluxes from both parts of the control volume have to be summed up correctly.

The mismatch between the centroid of the control volume and the node where the residual is stored has also a further implication for the median-dual scheme. It arises as the mass matrix in the case of unsteady flows. We discussed this point already at the beginning of Section 3.2. The advantage of the cell-centred scheme is that the mass matrix can be eliminated from the equations, without compromising the solution accuracy. By contrast, the median-dual scheme requires a special treatment of the mass matrix [41], [42].

In order to judge the computational effort required for both schemes, we have to consider primarily the integration of the fluxes. We know from the previous discussion that the cell-centred scheme uses a loop over cell faces whereas the median-dual scheme loops over the edges. Since the evaluation of the fluxes at an interface is quite similar for both schemes, the ratio of the number of cell faces to the number of edges gives the ratio of the computational work. Thus, on a tetrahedral grid, where the number cell faces (if counted only once for every two cells) is approximately two times larger than the number of edges, the cell-centred scheme is computationally twice as much expensive as the median-dual scheme on an identical grid [36]. The cell-centred approach becomes however more competitive on mixed grids containing prismatic elements. Apart from boundary treatment, both methods are computationally equivalent on hexahedral grids, where the number of faces equals the number of edges.

Considering the memory requirements, the cell-centred scheme has to store about six times more flow variables on tetrahedral and about three times more variables on usual mixed grids as compared to the median-dual scheme. Furthermore, as we saw, both schemes require to store two integers and three reals (pointers and face vector) per cell face or edge, respectively. Additionally, the cell-centred scheme has to keep two vectors to the face-midpoint - 6 reals - per cell face in memory. On the contrary, the median-dual scheme can work with the node coordinates only, which are considerably fewer values. Thus in summary, the cell-centred scheme needs, on average, more than twice as much computer memory as the median-dual method.

One significant advantage of the cell-centred scheme appears in the case of non-conforming cell interfaces, like those at the letter "F" in Fig. 3.4. In contrast to the median-dual methodology, no special and expensive procedure is required for the computation of the fluxes at the interface. This allows for an increased flexibility in the grid generation and also in the grid adaptation.