6.1.2 Hybrid Multistage Schemes 混合多步格式[cfd-6-1-2]

把显式多步格式(6.5)应用于方程组(6.1)时ï¼如果黏性通量和耗散不在每个阶段都重新求值ï¼计算量可以大幅减少。此外ï¼还可以把不同阶段的耗散项混合(blend)起来ï¼以提高格式的稳定性。这类方法由Martinelli[5]和Mavriplis等人[6]提出ï¼称为混合多步格式(hybrid multistage schemes)。只要阶段系数经过仔细优化ï¼混合格式与基本多步格式一样稳健。en

The computational work of an explicit multistage scheme (6.5), applied to the system (6.1), can be substantially reduced if the viscous fluxes and the dissipation are not re-evaluated at each stage. Additionally, the dissipation terms from different stages can be blended to increase the stability of the scheme. Methods of this type were devised by Martinelli [5] and by Mavriplis et al. [6]. They are known as hybrid multistage schemes. Provided the stage coefficients are carefully optimised, the hybrid schemes are as robust as the basic multistage schemes.

为便于说明ï¼考虑一种常用的5步混合格式ï¼其耗散项只在奇数步求值——一般记为(5,3)格式。首先ï¼把空间离散化分成两部分ï¼即en

Let us for illustration consider a popular 5-stage hybrid scheme, where the dissipative terms are evaluated at odd stages - generally denoted as the (5,3)-scheme. First, we split the spatial discretisation into two parts, i.e.,

\vec{R}_I = \left(\vec{R}_c\right)_I - \left(\vec{R}_d\right)_I. \tag{6.6}

第一部分 $\vec{R}_c$ 包含对流通量的中心离散化——可以是变量平均ï¼也可以是通量平均——还包括源项。第二部分 $\vec{R}_d$ 由黏性通量和数值耗散组成。例如ï¼对于带人工耗散的中心格式(4.3.1或5.3.1小节)ï¼可取en

The first part, $\vec{R}_c$ , contains the central discretisation of the convective fluxes, which can be either the average of variables or the average of fluxes. It also includes the source term. The second part, $\vec{R}_d$ , is composed of the viscous fluxes and the numerical dissipation. For example, in the case of the central scheme with artificial dissipation (Subsections 4.3.1 or 5.3.1) we would set

\begin{aligned}\left(\vec{R}_c\right)_I &= \sum_{k=1}^{N_F}\left[\vec{F}_c(\vec{W}_{av})\Delta S\right]_k - \left(\vec{Q}\Omega\right)_I \\\left(\vec{R}_d\right)_I &= \sum_{k=1}^{N_F}\left[\vec{F}_v\Delta S + \vec{D}\right]_k,\end{aligned} \tag{2}

其中 $\vec{W}_{av}$ 表示面 $$k$$ 左右两侧流动变量的算术平均。en

where $\vec{W}_{av}$ represents the arithmetic average of flow variables from the left and the right side of face $$k$$ .

残差按方程(6.6)分裂之后,(5,3)格式可以写为en

With the residual split according to Eq. (6.6), the (5,3)-scheme can be formulated as

\begin{aligned}\vec{W}_I^{(0)} &= \vec{W}_I^n \\\vec{W}_I^{(1)} &= \vec{W}_I^{(0)} - \alpha_1\frac{\Delta t_I}{\Omega_I}\left[\vec{R}_c^{(0)} - \vec{R}_d^{(0)}\right]_I \\\vec{W}_I^{(2)} &= \vec{W}_I^{(0)} - \alpha_2\frac{\Delta t_I}{\Omega_I}\left[\vec{R}_c^{(1)} - \vec{R}_d^{(0)}\right]_I \\\vec{W}_I^{(3)} &= \vec{W}_I^{(0)} - \alpha_3\frac{\Delta t_I}{\Omega_I}\left[\vec{R}_c^{(2)} - \vec{R}_d^{(2,0)}\right]_I \\\vec{W}_I^{(4)} &= \vec{W}_I^{(0)} - \alpha_4\frac{\Delta t_I}{\Omega_I}\left[\vec{R}_c^{(3)} - \vec{R}_d^{(2,0)}\right]_I \\\vec{W}_I^{n+1} &= \vec{W}_I^{(0)} - \alpha_5\frac{\Delta t_I}{\Omega_I}\left[\vec{R}_c^{(4)} - \vec{R}_d^{(4,2)}\right]_I,\end{aligned} \tag{6.7}

其中en

where

\begin{aligned}\vec{R}_d^{(2,0)} &= \beta_3\vec{R}_d^{(2)} + \left(1-\beta_3\right)\vec{R}_d^{(0)} \\\vec{R}_d^{(4,2)} &= \beta_5\vec{R}_d^{(4)} + \left(1-\beta_5\right)\vec{R}_d^{(2,0)}.\end{aligned} \tag{6.8}

上述关系(6.7)、(6.8)中的阶段系数 $\alpha_m$ 和混合系数 $\beta_m$ 在表6.2中针对中心格式和上风格式给出。这两组系数都是专门针对多重网格方法(9.4节)优化的。上述混合多步格式的性质将在10.3节中讨论。en

The stage coefficients $\alpha_m$ and the blending coefficients $\beta_m$ in the above relations (6.7), (6.8) are given in Table 6.2 for central and upwind schemes. Both sets of coefficients are particularly optimised for the multigrid method (Section 9.4). We shall discuss the properties of the above hybrid multistage scheme later in Section 10.3.

应当指出ï¼另一种常见的做法是只在头两步求值耗散项 $\vec{R}_d$ ï¼而不做任何混合。著名的(5,2)格式常与中心空间离散化配合使用ï¼它采用表6.2中的阶段系数。然而ï¼与上述(5,3)格式相比,(5,2)格式不太适合黏性流动和多重网格。en

It should be mentioned that it is also popular to evaluate the dissipation term $\vec{R}_d$ in the first two stages only, without any blending. A well-known (5,2)-scheme, which is often employed with the central spatial discretisation, uses the stage coefficients of Table 6.2. However, the (5,2)-scheme is less suitable for viscous flows and multigrid than the above (5,3)-scheme.