9.3.3 Upwind IRS on Structured Grids 结构网格上的上风IRS[cfd-9-3-3]

前面讨论的CIRS方法对亚声速和跨声速流动工作良好，在黏性占主导的区域也很有帮助。然而，与上风空间离散格式结合使用时，CIRS的误差阻尼特性较差。此外，经CIRS加速的多级格式在强激波情形下鲁棒性会下降。为此，人们发展了所谓的上风隐式残差光顺（Upwind Implicit Residual Smoothing，UIRS）方法[15]、[16]，它特别适合高马赫数流动。en

The previously discussed CIRS method works satisfactorily for subsonic and transonic flows. It is also helpful in viscous dominated regions. However, CIRS exhibits poor error-damping characteristics in conjunction with upwind spatial discretisation schemes. Furthermore, the robustness of a multistage scheme accelerated by CIRS suffers in the case of strong shocks. Therefore, the so-called Upwind Implicit Residual Smoothing (UIRS) method was developed [15], [16], which is particularly suited for high Mach-number flows.

与CIRS不同，UIRS方法考虑了对流特征值 $\bar{\Lambda}_c$ 的符号。其思想是只在欧拉方程特征线的方向上光顺残差（即 $dx/dt = \mathrm{const.} = \Lambda_c$ ）。这种做法可防止对上游残差产生非物理的影响。UIRS方法要求把残差变换到特征变量（参见附录A.11）。这样，残差向量的每个分量都可以按照相应特征值的符号独立地进行光顺。对于残差的第 $$l$$ 个分量，一维情形下的隐式算子定义为[15]、[16]en

By contrast to CIRS, the UIRS methodology takes the sign of the convective eigenvalues $\bar{\Lambda}_c$ into account. The idea is to smooth the residuals only in the direction of the characteristic of the Euler equations (i.e., $dx/dt = \mathrm{const.} = \Lambda_c$ ). This approach prevents unphysical influences on the upstream residuals. The UIRS method requires transformation of the residuals into the characteristic variables (cf. Appendix A.11). In this way, each component of the residual vector can be smoothed independently according to the sign of the corresponding eigenvalue. For the $$l$$ -th component of the residual, the implicit operator is defined in 1D as [15], [16]

\begin{aligned} -\epsilon^I\left(R^{*}\right)^l_{I-1} + \left(1 + \epsilon^I\right)\left(R^{*}\right)^l_I \qquad &= \left(R^{c}\right)^l_I \quad\text{if}\quad \Lambda_c^l \ge 0\\ \left(1 + \epsilon^I\right)\left(R^{*}\right)^l_I - \epsilon^I\left(R^{*}\right)^l_{I+1} &= \left(R^{c}\right)^l_I \quad\text{if}\quad \Lambda_c^l < 0\ , \end{aligned} \tag{9.11}

其中 $\vec{R}^{c}$ 表示变换到特征变量后的残差，即en

where $\vec{R}^{c}$ denotes the residual transformed into characteristic variables, i.e.,

\vec{R}^{c} = \tilde{T}^{-1}\vec{R}\,. \tag{9.12}

光顺后的残差 $\vec{R}^{*}$ 通过用Thomas算法求解一个三对角（若 $\Lambda_c^l$ 不变号则为二对角）方程组而得到。随后，把残差 $\vec{R}^{*}$ 变换回守恒变量，之后即可更新解。en

The smoothed residuals $\vec{R}^{*}$ are obtained by the solution of a tridiagonal (bidiagonal if $\Lambda_c^l$ does not change its sign) equation system using the Thomas algorithm. Afterwards, the residuals $\vec{R}^{*}$ are transformed back into the conservative variables and the solution can be updated.

UIRS最理想的特性在于，它使多级格式具有非常好的阻尼特性，与上风空间离散格式结合时尤其如此。在光顺系数很大时，阻尼解误差的能力仍保持不变甚至有所提高。对一维欧拉方程已经证明，像 $\epsilon = 500$ 和 $\sigma^{*} = 1000$ 这样的取值可以得到稳定且非常快的显式多级格式[15]、[16]（另见CD-ROM中的analysis/mstage）。然而，问题在于UIRS在多维情形下的实现。由于对流通量雅可比矩阵无法在所有坐标方向上同时对角化（见附录A.11），变换式(9.12)与光顺式(9.11)必须对每个计算坐标分别执行。坐标分裂的后果是最大光顺系数降为 $2 \le \epsilon \le 6$ 。尽管如此，与CIRS相比，向定常态的收敛仍被大大加速[17]、[16]。在收敛速度与鲁棒性方面最大的改进，出现在与多重网格联合使用时[18]、[16]。en

The most desirable feature of UIRS is that it leads to very favourable damping properties of the multistage scheme, particularly in connection with an upwind spatial discretisation. The ability to damp solution errors remains or even improves for high smoothing coefficients. It was demonstrated for 1-D Euler equations that values like $\epsilon = 500$ and $\sigma^{*} = 1000$ result in a stable and very fast explicit multistage scheme [15], [16] (see also in analysis/mstage on the CD-ROM). However, the problem is the implementation of UIRS in multiple dimensions. Since the convective flux Jacobians cannot be diagonalised simultaneously in all coordinate directions (see Appendix A.11), the transformation Eq. (9.12) and the smoothing Eq. (9.11) have to be carried out separately for each computational coordinate. The effect of the coordinate splitting is a reduced maximum smoothing coefficient to $2 \le \epsilon \le 6$ . Despite this, the convergence to the steady state is strongly accelerated as compared to CIRS [17], [16]. The largest improvements in terms of convergence speed and robustness occur in combination with multigrid [18], [16].

多维情形下的光顺系数按特征值进行缩放。例如在二维情形下，可以采用如下公式en

The smoothing coefficients in multiple dimensions are scaled by the eigenvalues. For example, in 2D the following formula can be employed

\begin{aligned} \epsilon^I &= \epsilon\cdot\min\left[\frac{\Lambda_c^I}{\Lambda_c^J},\,1\right]\\ \epsilon^J &= \epsilon\cdot\min\left[\frac{\Lambda_c^J}{\Lambda_c^I},\,1\right]\ . \end{aligned} \tag{9.13}

光顺格式的CFL数与系数 $\epsilon$ 之间的关系为en

The relation between the CFL number of the smoothed scheme and the coefficient $\epsilon$ reads

\frac{\sigma^{*}}{\sigma} \le 1 + C\epsilon\,. \tag{9.14}

常数 $$C$$ 取决于空间离散格式的类型。中心格式为 $$C = 1$$ ；一阶或二阶上风格式则为 $$C = 2$$ 。en

The constant $$C$$ depends on the kind of the spatial discretisation. In the case of the central scheme $$C = 1$$ . For the 1st- or 2nd-order upwind scheme the value is $$C = 2$$ .

为了避免变换到特征变量所需的计算量，有人提出了UIRS方法的简化版本[17]、[18]、[16]。按 $$I$$ 方向写出即为en

In order to circumvent the numerical effort of the transformation to the characteristic variables, a simplified version of the UIRS method was suggested [17], [18], [16]. Written in the $$I$$ -direction it reads

\begin{aligned} -\epsilon^I\vec{R}^{*}_{I-1} + \left(1 + \epsilon^I\right)\vec{R}^{*}_{I} \qquad\qquad &= \vec{R}_{I} \quad\text{if}\quad M^I > 1\\ -\epsilon^I\vec{R}^{*}_{I-1} + \left(1 + 2\epsilon^I\right)\vec{R}^{*}_{I} - \epsilon^I\vec{R}^{*}_{I+1} &= \vec{R}_{I} \quad\text{if}\quad \left|M^I\right| \le 1\\ \left(1 + \epsilon^I\right)\vec{R}^{*}_{I} - \epsilon^I\vec{R}^{*}_{I+1} \;\; &= \vec{R}_{I} \quad\text{if}\quad M^I < -1\ . \end{aligned} \tag{9.15}

马赫数 $$M$$ 基于投影到相应计算坐标方向（这里为 $$I$$ ）上的速度。由于运算量低，简化UIRS特别适合三维流动问题。文献[16]中证明，与CIRS相比，到达定常态所需的CPU时间可以减半（绕钝头圆柱的高超声速流动， $M_\infty = 8$ ）。en

The Mach number $$M$$ is based on velocity projected into the direction of the particular computational coordinate (here: $$I$$ ). Due to the low operation count, the simplified UIRS is especially suitable for 3-D flow problems. In Ref. [16] it was demonstrated that the CPU time needed to reach the steady state can be halved as compared to CIRS (hypersonic flow past a blunt cylinder, $M_\infty = 8$ ).