9.4.3 Implementation on Structured Grids 结构网格上的实现[cfd-9-4-3]

The implementation of multigrid on structured grids is straightforward since the coarse grids can be easily generated by deleting every second grid line in the respective coordinate direction. The spacing of the grid lines is therefore \(2h\), \(4h\), etc. This guarantees that the numerical effort on the coarse grids stays low as compared to the finest grid. Several representative examples are provided in Refs. [11]-[13], [20]-[22], [26], [32]-[36].

As we can conclude from Fig. 9.5, the operators for the solution interpolation, the restriction of residuals and the prolongation of corrections have to be defined differently for cell-centred and node-centred (cell-vertex) schemes. For example, two successive grids have every second grid point in common. On the contrary, the cell centres are always at different locations. Therefore, we shall discuss the standard forms of the transfer operators separately for the cell-centred and node-centred (identical to cell-vertex) finite-volume schemes.

Apart from the symmetrical, purely geometrically defined restriction and prolongation operators, which will be presented next, upwind-biased forms were suggested in [16], Chapter 4. Upwind restriction and prolongation, which account both for the characteristics of the flow equations, improve the robustness of the multigrid scheme for hypersonic flows. In order to save space, we shall discuss the implementation of an upwind prolongation operator for the node-centred spatial discretisation only.

The solution is transfered from the fine to the coarse grid by using a volume weighted interpolation. In 2D, Eq. (9.17) becomes (see Fig. 9.6a)

Similar transfer operator is employed in 3D, where the summation is over the eight fine-grid control volumes, which form one coarse-grid cell.

The restriction operator is defined as a sum of the residuals from all cells which are contained in one coarse-grid control volume. Hence, in 2D we have (cf. Fig. 9.6a)

and likewise in 3D. Those residuals \(\vec{R}^{n+1}_{h}\) in Eq. (9.27), which are located outside the physical domain, have to be set to zero.

The prolongation of the coarse-grid correction Eq. (9.23) can be conducted in two different ways. First, a zeroth-order prolongation operator results, if \(\delta\vec{W}_{2h}\) is equally distributed to all surrounding cell centres as indicated in Fig. 9.6b. Thus, for instance

The second possibility, which leads to a faster convergence of the multigrid scheme, consists of two steps. In a first step, \(\delta\vec{W}_{2h}\) is interpolated to the grid nodes like for the cell-vertex scheme (see below). In a second step, the nodal values are averaged to obtain the value in the centre of the fine-grid cell. Referring to Fig. 9.6c, the following final relationship can be derived

The corresponding expression in 3D can be found by a similar procedure. It reads

Because of the common nodes between the fine and coarse grid, the solution can be transfered simply by injection, i.e.,

The standard central restriction operator represents a linear interpolation from the nodes of all four (eight in 3D) fine-grid cells, which resemble one coarse-grid cell. According to Fig. 9.7, the restricted residual is computed in 2D as

In 3D, the fine-grid residuals are collected as follows

with the factors

In the above Eq. (9.34), only those indices are shown which are different from \(i, j, k\). It should be noted that the residuals \(\vec{R}^{n+1}_{h}\) must be set to zero at all physical boundary and dummy points before the restriction.

The prolongation of the coarse-grid correction can be implemented as a loop over the points of the coarse grid. Within the loop, the values \((\delta\vec{W}_{2h})_{i,j,k}\) are distributed to the fine-grid points using the same weights as for the restriction (cf. Fig. 9.7b). The particular contributions are summed up in order to obtain the complete transfered correction at each point of the fine grid.

In principle, upwind prolongation can be formulated either in characteristic or in conservative variables. A particularly efficient implementation in conservative variables was proposed in [16]. The methodology employs upwind-biased interpolation of the corrections in Eq. (9.23) according to the Mach number and the velocity direction. The numerical effort is very low, nevertheless the robustness of the multigrid scheme can be significantly improved for high Mach-number flows [16].

The interpolation of the solution corrections \(\delta\vec{W}_{2h}\) to the finer grid is accomplished in two steps. In the first step, the corrections at the points \(A\), \(B\), \(C\), and \(D\) (see Fig. 9.8), which are common to both grids, are transferred directly to the finer grid. In the second step, the corrections are interpolated to the points \(e\), \(f\), \(e'\), \(f'\) and \(g\), which are contained only on the finer grid. The interpolation depends on the sign and the absolute value of the Mach number at the corresponding point. The Mach number in this case is calculated using

is employed. The normal vector \(\vec{n}\) in Eq. (9.35) can be obtained either by averaging the face vectors of the control volume (in the direction \(A-B\)), or by normalising the vector from point \(A\) to point \(B\). Then, the correction is transferred to point \(e\) according to the rule

The same procedure applies also to the points \(f\) to \(f'\). The upwinding helps to match the information exchange between the grids better to the real physics. The interpolation to the point \(g\) is more difficult. In Ref. [16], the values at the surrounding points \(A\) to \(D\) were simply averaged

However, some sort of upwind weighted interpolation would be more appropriate. The upwind prolongation can be implemented in similar way also in 3D. Despite the simplification, encouraging results were obtained in a number of test cases [16].