6.2.3 ADI Scheme ADI格式[cfd-6-2-3]

交替方向隐式(Alternating Direction Implicit,ADI)格式是最早的迭代式隐式格式之一[40]。ADI格式只能在结构网格上实现。它基于把式(6.28)中的隐式算子近似分裂(或者说ï¼因式分解)为二维时的两个、三维时的三个因子。每个因子包含计算空间中某一特定方向的对流通量与黏性通量的线性化。在三维情形ï¼这导出如下形式[40]、[41]en

The Alternating Direction Implicit (ADI) scheme was one of the first iterative implicit schemes [40]. The ADI scheme can be implemented on structured grids only. It is based on an approximate splitting (or, in other words, factorisation) of the implicit operator in Eq. (6.28) into two (in 2D) or three (in 3D) factors. Each factor contains the linearisation of the convective and viscous fluxes for one particular direction in the computational space. In 3D, this leads to the formulation [40], [41]

\begin{gathered} \left\{ \frac{\Omega}{\Delta t}\bar{I} + \frac{\partial\left[\left(\vec{F}^{I}_{c} - \vec{F}^{I}_{v}\right)\Delta S^{I}\right]_{i+1/2}}{\partial\vec{W}} + \frac{\partial\left[\left(\vec{F}^{I}_{c} - \vec{F}^{I}_{v}\right)\Delta S^{I}\right]_{i-1/2}}{\partial\vec{W}} \right\}\\ \left\{ \frac{\Omega}{\Delta t}\bar{I} + \frac{\partial\left[\left(\vec{F}^{J}_{c} - \vec{F}^{J}_{v}\right)\Delta S^{J}\right]_{j+1/2}}{\partial\vec{W}} + \frac{\partial\left[\left(\vec{F}^{J}_{c} - \vec{F}^{J}_{v}\right)\Delta S^{J}\right]_{j-1/2}}{\partial\vec{W}} \right\}\\ \left\{ \frac{\Omega}{\Delta t}\bar{I} + \frac{\partial\left[\left(\vec{F}^{K}_{c} - \vec{F}^{K}_{v}\right)\Delta S^{K}\right]_{k+1/2}}{\partial\vec{W}} + \frac{\partial\left[\left(\vec{F}^{K}_{c} - \vec{F}^{K}_{v}\right)\Delta S^{K}\right]_{k-1/2}}{\partial\vec{W}} - \frac{\partial(\Omega\vec{Q})}{\partial\vec{W}} \right\}\Delta\vec{W}^n = -\vec{R}^n. \end{gathered} \tag{6.46}

为清晰起见ï¼式(6.46)中在不必要之处省略了节点指标 $$i$$ 、 $$j$$ 、 $$k$$ 。此外ï¼式(6.46)中的上标 $$I$$ 、 $$J$$ 、 $$K$$ 标记与计算空间中某一坐标相关联的通量向量或面面积。例如, $\Delta S^{I}_{i+1/2,j,k}$ 对应于图4.1b中的 $\Delta S_2$ 。把节点指标换成单元指标ï¼式(6.46)的格式即可用于单元中心离散。en

For clarity, the node indices $$i, j, k$$ were omitted from Eq. (6.46) where not required. Furthermore, the superscripts $$I, J, K$$ in Eq. (6.46) mark the flux vector or of the face area associated with certain coordinate in the computational space. For example, $\Delta S^{I}_{i+1/2,j,k}$ corresponds to $\Delta S_2$ in Fig. 4.1b. By replacing the node indices by cell indices, the scheme in Eq. (6.46) can be applied to a cell-centred discretisation.

式(6.46)中对流通量与黏性通量的导数可以按上一小节所讨论的方法计算。ADI方法传统上与带人工耗散的中心格式耦合使用。在这种情形下ï¼每个因子都具有与式(6.34)类似的形式(当然ï¼源项的线性化只包含在一个因子中)。为了得到稳健而高效的格式ï¼必须在隐式算子中纳入人工耗散项的线性化[42]-[44]。该线性化一般通过把谱半径与耗散系数视为与 $\vec{W}$ 无关来简化。于是ï¼根据式(4.48)ï¼例如I方向上的因子变为en

The derivatives of the convective and viscous fluxes in Eq. (6.46) can be evaluated as discussed in the previous subsection. The ADI method is traditionally coupled to the central scheme with artificial dissipation. In such a case, a formulation similar to that in Eq. (6.34) holds for each factor (of course, the linearisation of the source term is included in one factor only). In order to obtain a robust and efficient scheme, it is necessary to include a linearisation of the artificial dissipation terms in the implicit operator [42]-[44]. The linearisation is in general simplified by treating the spectral radii and the dissipation coefficients as independent of $\vec{W}$ . Hence, according to Eq. (4.48) the factor, e.g., in the I-direction becomes

\left\{ \frac{\Omega}{\Delta t}\bar{I} + \frac{\partial\left[\left(\vec{F}^{I}_{c} - \vec{F}^{I}_{v}\right)\Delta S^{I}\right]_{i+1/2}}{\partial\vec{W}} + \frac{\partial\left[\left(\vec{F}^{I}_{c} - \vec{F}^{I}_{v}\right)\Delta S^{I}\right]_{i-1/2}}{\partial\vec{W}} - \frac{\partial\left(\vec{D}^{I}_{IM}\Delta S^{I}\right)_{i+1/2}}{\partial\vec{W}} - \frac{\partial\left(\vec{D}^{I}_{IM}\Delta S^{I}\right)_{i-1/2}}{\partial\vec{W}} \right\} \tag{2}

其中的隐式人工耗散项按JST格式(参见式(4.50))为en

with the implicit artificial dissipation term JST scheme, (cf. Eq. (4.50))

\frac{\partial\left(\vec{D}^{I}_{IM}\right)_{i+1/2}}{\partial\vec{W}}\Delta\vec{W}^n \approx \hat{\Lambda}^{S}_{i+1/2}\left[\left(\epsilon^{(2)}_{IM}\right)_{i+1/2}\left(\Delta\vec{W}^{n}_{i+1} - \Delta\vec{W}^{n}_{i}\right) - \left(\epsilon^{(4)}_{IM}\right)_{i+1/2}\left(\Delta\vec{W}^{n}_{i+2} - 3\Delta\vec{W}^{n}_{i+1} + 3\Delta\vec{W}^{n}_{i} - \Delta\vec{W}^{n}_{i-1}\right)\right]. \tag{6.47}

其他方向的隐式耗散项按类似方式定义。也可以在隐式算子中只保留二阶差分[43]、[44]。这会把每个因子的矩阵从块五对角形式约化为块三对角形式(如图6.1所示意)。然而ï¼正如文献[44]所指出的ï¼这会限制格式的稳定性。en

The implicit dissipation terms in other directions are defined in similar way. It is also possible to retain only the second-order differences in the implicit operator [43], [44]. This reduces the matrix for each of the factors from block-pentadiagonal to block-tridiagonal form (as sketched in Fig. 6.1). However, as pointed out in Ref. [44], this restricts the stability of the scheme.

式(6.46)中隐式算子的求逆分三步进行(二维时为两步)ï¼即en

The inversion of the implicit operator in Eq. (6.46) proceeds in three steps (two steps in 2D), i.e.,

\begin{aligned} \left(\mathbf{D}^{I} + \mathbf{L}^{I} + \mathbf{U}^{I}\right)\Delta\vec{W}^{(1)} &= -\vec{R}^n\\ \left(\mathbf{D}^{J} + \mathbf{L}^{J} + \mathbf{U}^{J}\right)\Delta\vec{W}^{(2)} &= \Delta\vec{W}^{(1)}\\ \left(\mathbf{D}^{K} + \mathbf{L}^{K} + \mathbf{U}^{K}\right)\Delta\vec{W}^{n} &= \Delta\vec{W}^{(2)}, \end{aligned} \tag{6.48}

其中 $\mathbf{D}$ 表示对角项, $\mathbf{L}$ 表示下对角项, $\mathbf{U}$ 表示上对角项。每一步都需要对块三对角或块五对角矩阵求逆(若式(6.47)中的 $\epsilon^{(4)}_{IM} > 0$ )。这由直接解法完成。为了减少数值工作量,Pulliam与Chaussee[45]提出了ADI格式的对角化形式。由此ï¼块矩阵(由对流通量与黏性通量雅可比组成)被变换为对角矩阵。于是只需对非块的三对角或五对角矩阵求逆ï¼可显著节省计算量与内存[43]。虽然这种对角化严格来说只对欧拉方程成立ï¼但它同样可用于黏性流动[43]。此时ï¼黏性通量的线性化(例如式(6.45)那样)要么在隐式算子中省略ï¼要么可以用黏性特征值(式(6.19))来近似。en

where $\mathbf{D}$ represents the diagonal, $\mathbf{L}$ the lower-diagonal and $\mathbf{U}$ the upper-diagonal terms, respectively. Each step requires the inversion of a block-tridiagonal or a block-pentadiagonal matrix (if $\epsilon^{(4)}_{IM} > 0$ in Eq. (6.47)). This is done by a direct solution method. In order to reduce the numerical effort, Pulliam and Chaussee [45] suggested a diagonalised form of the ADI scheme. Herewith, the block matrices (composed of convective and viscous flux Jacobians) are transformed into diagonal matrices. Hence, only non-block tri- or pentadiagonal matrices have to be inverted, which results in significant savings of computational work and memory [43]. Although the diagonalisation is strictly valid only for Euler equations, it can be employed for viscous flows as well [43]. The linearisation of the viscous fluxes (e.g., like in Eq. (6.45)) is then either omitted in the implicit operator, or it can be approximated by the viscous eigenvalue (Eq. (6.19)).

对隐式算子的分裂引入了所谓的因式分解误差(factorisation error)。它是式(6.28)基本格式的隐式算子与因式分解后算子之差。对ADI格式而言ï¼该误差项以因子 $(\Delta t)^{N}$ 缩放ï¼其中 $$N$$ 为空间维数。这一项使ADI格式在三维中失去无条件稳定性[46]。不过ï¼如果把人工黏性的四阶差分包含在隐式算子中ï¼稳定性会得到改善[44]。事实上,ADI格式已被成功地用于求解各种三维问题[43]、[47]。Rosenfeld等人[48]提出了一种在多块网格上隐式处理块边界的ADI格式的有趣实现。en

The splitting of the implicit operator introduces what is called the factorisation error. It is the difference between the implicit operator of the base scheme in Eq. (6.28) and the factorised operator. In the case of the ADI scheme, this error term is scaled by the factor $(\Delta t)^{N}$ , where $$N$$ denotes the number of space dimensions. This term causes the ADI scheme to loose its unconditional stability in 3D [46]. However, the stability is improved if the fourth-order differences of the artificial viscosity are included in the implicit operator [44]. In fact, the ADI scheme was successfully used for the solution of various 3-D problems [43], [47]. An interesting implementation of the ADI scheme on multiblock grids, which treats the block boundaries implicitly, was presented by Rosenfeld et al. [48].

时间步长 $\Delta t$ 可以按6.1.4小节所述方法、用式(6.14)计算。最优CFL数依流动工况在20到50之间变化。en

The time step $\Delta t$ can be computed in the same way as presented in Subsection 6.1.4, using Eq. (6.14). The optimal CFL number varies between 20 and 50 depending on the flow case.