9.4.2 Multigrid Strategies 多重网格策略[cfd-9-4-2]

上面描述的基本多重网格格式只含一个粗网格。如果存在多个粗网格，则重复步骤1和步骤2，直到到达最粗网格。重要的是要认识到，粗网格上的强迫函数是由式(9.19)限制后的修正残差构成的。例如，在粗网格 $$4h$$ 上，强迫函数由下式得到en

The basic multigrid scheme described above consists of one coarse grid only. If multiple coarse grids are present, steps 1 and 2 are repeated until the coarsest grid is reached. It is important to realize that the forcing function on the coarse grids is formed from the restricted corrected residual of Eq. (9.19)). For example, on the coarse grid $$4h$$ , the forcing function is obtained from

(\vec{Q}_F)_{4h} = I^{4h}_{2h}(\vec{R}_F)^{n+1}_{2h} - \vec{R}^{(0)}_{4h} = I^{4h}_{2h}\left[\vec{R}^{n+1}_{2h} + (\vec{Q}_F)_{2h}\right] - \vec{R}^{(0)}_{4h}\,. \tag{9.24}

这样，最细网格的残差控制着所有粗网格上解的精度。在最粗网格上进行给定数目的时间步之后，可以逐层重复步骤3，直到再次到达最细网格。这一过程称为锯齿形循环或V循环（见图9.4a）。不过，也可以在粗网格上执行更多的循环。这种策略称为W循环，如图9.4b所示。它在跨声速流动中采用得尤其频繁。而对于超声速和高超声速流动，V循环被证明效率更高。en

In this way, the residual of the finest grid controls the accuracy of the solution on all coarse grids. After a given number of time steps on the coarsest grid, step 3 can be successively repeated until the finest grid is reached again. This procedure is known as a saw-tooth or V-cycle (see Fig. 9.4a). However, it is also possible to conduct more cycles on the coarse grids. This strategy, termed the W-cycle, is displayed in Fig. 9.4b. It is employed particularly frequently for transonic flows. In the case of supersonic and hypersonic flows, the V-cycle proved to be more efficient.

Number of Time Steps 时间步数

限制前和延拓后的最优时间步数取决于时间推进格式的类型。对于显式多级格式（6.1.1或6.1.2小节），通常在残差限制之前只进行一个时间步，延拓之后不进行时间步。不过，把粗网格修正（式(9.22)）在加到细网格解 $\vec{W}^{n+1}_{h}$ （式(9.23)）上之前先做光顺，可以改进多重网格格式的鲁棒性。这里采用与9.3节相同的中心隐式光顺（系数取为常数）。en

The optimum number of time steps before the restriction and after the prolongation depends on the type of the time-stepping scheme. In the case of the explicit multistage scheme (Subsection 6.1.1 or 6.1.2), it is common to carry out only one time step before the restriction of residuals and no time step after the prolongation. However, the robustness of the multigrid scheme can be improved by smoothing the coarse grid corrections (Eq. (9.22)) before adding them to the fine grid solution $\vec{W}^{n+1}_{h}$ (Eq. (9.23)). The same central implicit smoothing (with constant coefficients) as described in Section 9.3 is utilised.

另一种常用的时间推进方法——隐式LU-SGS格式（见6.2.4小节）——为了获得最佳多重网格效率，需要在限制之前进行两次迭代[34]。延拓之后的时间步数则取决于空间离散。对于中心格式（4.3.1小节），不需要时间步[34]-[36]，但可以对解修正进行光顺。相反，如果采用上风空间离散，则应在延拓之后进行一个时间步。实践证明，这种(2,1)策略在各种流动条件下的鲁棒性和计算时间方面都是最优的[35]、[36]。en

The other popular time-stepping method, the implicit LU-SGS scheme (see Subsection 6.2.4), requires two iterations before the restriction for the best multigrid efficiency [34]. The number of time steps after the prolongation depends on the spatial discretisation. In the case of the central scheme (Subsection 4.3.1), no time step is necessary [34]-[36], but the solution correction can be smoothed. On the contrary, one time step should carried out after the prolongation if an upwind spatial discretisation is used. This (2,1)-strategy proved to be an optimum with respect to robustness and computing time for various flow conditions [35], [36].

Starting Grid 起始网格

需要指出的是，实际中多重网格格式并不是直接从最细网格开始的。相反，先从某个粗网格开始执行若干个多重网格循环，把近似解插值到下一层较细的网格（采用与延拓相同的算子），再执行几个循环，然后把解再次插值到下一层更细的网格，如此继续，直到最细网格。这样，只需适度的数值工作量，就能在最细网格上获得一个良好的起始解。这一非常高效的过程称为完全多重网格（Full Multigrid，FMG）方法[19]。en

It should be pointed out that in practice the multigrid scheme is not started directly from the finest grid. Instead, several multigrid cycles are executed from one of the coarse grids. The approximate solution is interpolated to the next finer grid (using the same operator as for the prolongation), few more cycles are performed, the solution is again interpolated to the next finer grid and so on, until the finest grid is reached. In this way, a good starting solution is obtained on the finest grid with only a moderate numerical effort. This very efficient procedure is termed the Full Multigrid (FMG) method [19].

图9.4:多重网格循环的类型

图9.4:多重网格循环的类型。图内标注:(a)V-cycle——V循环;(b)W-cycle——W循环;h、2h、4h、8h——网格层(由细到粗);●——限制前的时间步;∘——延拓后的时间步。

Accuracy of Transfer Operators 转移算子的精度

限制算子(9.18)与延拓算子(9.23)必须满足一定的精度要求，即[52]en

The restriction (9.18) and the prolongation (9.23) operator must fulfil certain accuracy requirements, namely [52]

m_R + m_P > m_E\,, \tag{9.25}

其中 $$m_R$$ 和 $$m_P$$ 分别表示限制算子和延拓算子能够精确插值的多项式的“次数加1”。例如，线性插值时 $$m_R$$ 或 $$m_P$$ 等于2。此外， $$m_E$$ 表示控制方程的阶数。因此，欧拉方程的 $$m_E = 1$$ ，Navier-Stokes方程的 $$m_E = 2$$ 。如果违反条件(9.25)，限制和/或延拓引入的额外误差将干扰细网格解，这样的多重网格格式将收敛得非常缓慢，甚至发散。en

where $$m_R$$ and $$m_P$$ denote the degree plus 1 of the polynomial, which is exactly interpolated by the restriction and the prolongation operator, respectively. For example, $$m_R$$ or $$m_P$$ are equal to two in the case of linear interpolation. Furthermore, $$m_E$$ represents the order of the governing equations. Thus, $$m_E = 1$$ for the Euler equations, and $$m_E = 2$$ in the case of the Navier-Stokes equations. If the condition (9.25) is violated, the additional errors introduced by the restriction and/or prolongation will disturb the fine-grid solution. Hence, such multigrid scheme will converge only slowly or it will even diverge.