6.3.2 Dual Time-Stepping for Implicit Schemes 隐式格式的双时间步进[cfd-6-3-2]

用隐式格式在伪时间 $t^{*}$ 内求解式(6.68)ï¼其实现方式与6.2节所述相同。首先ï¼把式(6.68)写成非线性隐式格式ï¼即en

The implementation of an implicit scheme for the solution of Eq. (6.68) in pseudo time $t^{*}$ proceeds in the same way as outlined in Section 6.2. First of all, we formulate Eq. (6.68) as an nonlinear implicit scheme, i.e.,

\frac{\partial}{\partial t^{*}}\left(\Omega^{n+1}_{I}\vec{W}^{*}_{I}\right) = -\left(\vec{R}^{*}_{I}\right)^{l+1} \tag{6.75}

其中 $$(l+1)$$ 为新伪时间层。再次注意ï¼时间导数中没有 $\bar{M}$ 。式(6.69)定义的非定常残差可在伪时间内线性化如下en

with $$(l+1)$$ being the new pseudo-time level. Note again the absence of $\bar{M}$ in the time derivative. The unsteady residual, which is defined in Eq. (6.69), can be linearised in pseudo time as follows

\left(\vec{R}^{*}\right)^{l+1} \approx \left(\vec{R}^{*}\right)^{l} + \frac{\partial\vec{R}^{*}}{\partial\vec{W}^{*}}\Delta\vec{W}^{*}, \tag{6.76}

其中 $\Delta\vec{W}^{*} = \left(\vec{W}^{*}\right)^{l+1} - \left(\vec{W}^{*}\right)^{l}$ ï¼通量雅可比定义为en

where $\Delta\vec{W}^{*} = \left(\vec{W}^{*}\right)^{l+1} - \left(\vec{W}^{*}\right)^{l}$ and the flux Jacobian is defined as

\frac{\partial\vec{R}^{*}}{\partial\vec{W}^{*}} = \frac{\partial\vec{R}}{\partial\vec{W}} + \frac{3}{2\Delta t}\left(\Omega\bar{M}\right)^{n+1}. \tag{6.77}

把上述线性化代入式(6.75)ï¼便得到未分解的隐式格式[106]en

If we insert the above linearisation into Eq. (6.75), we obtain the unfactored implicit scheme [106]

\left[\left(\frac{1}{\Delta t^{*}_{I}} + \frac{3}{2\Delta t}\right)\left(\Omega\bar{M}\right)^{n+1}_{I} + \left(\frac{\partial\vec{R}}{\partial\vec{W}}\right)_{I}\right]\Delta\vec{W}^{*} = -\left(\vec{R}^{*}_{I}\right)^{l}. \tag{6.78}

6.2节介绍的任何方法都可用于求解方程组(6.78)。关于时间精确隐式方法的详细讨论可参见[107];近年实现的例子可参见例如[106]与[108]。en

Any of the methodologies presented in Section 6.2 can be employed for the solution of the system (6.78). A detailed discussion of time-accurate implicit methods can be found in [107]. For recent examples of implementations, the reader is referred to, e.g., [106] and [108].