6.1.1 Multistage Schemes (Runge-Kutta) 多步格式(Runge-Kutta)[cfd-6-1-1]

显式多步格式的概念最早由Jameson等人[1]提出。多步格式通过若干步——所谓阶段(stages)——来推进解ï¼每一步都可以看作按方程(6.4)进行的一次更新。将其应用于质量矩阵已集中的离散化控制方程(6.1), $$m$$ 步(m-stage)格式为en

The concept of explicit multistage schemes was first presented by Jameson et al. [1]. The multistage scheme advances the solution in a number of steps - so-called stages - which can be viewed as a sequence of updates according to Eq. (6.4). Applied to the discretised governing equations (6.1), where the mass matrix was lumped, an $$m$$ -stage scheme reads

\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}\vec{R}_I^{(0)} \\\vec{W}_I^{(2)} &= \vec{W}_I^{(0)} - \alpha_2\frac{\Delta t_I}{\Omega_I}\vec{R}_I^{(1)} \\&\;\;\vdots \\\vec{W}_I^{n+1} = \vec{W}_I^{(m)} &= \vec{W}_I^{(0)} - \alpha_m\frac{\Delta t_I}{\Omega_I}\vec{R}_I^{(m-1)}.\end{aligned} \tag{6.5}

在上面的表达式(6.5)中, $\alpha_k$ 表示阶段系数。此外ï¼记号 $\vec{R}_I^{(k)}$ 表示残差用第 $$k$$ 步的解 $\vec{W}_I^{(k)}$ 来求值。en

In the above expressions (6.5), $\alpha_k$ represents the stage coefficients. Furthermore, the denotation $\vec{R}_I^{(k)}$ means that the residual is evaluated with the solution $\vec{W}_I^{(k)}$ from the $$k$$ -th stage.

与经典Runge-Kutta格式不同ï¼这里为了减少内存需求ï¼只存储第零步的解和最新的残差。针对具体的空间离散化ï¼可以调整阶段系数ï¼以增大最大时间步长并改善稳定性[2]-[4]。为保持一致性ï¼只要求 $\alpha_m = 1$ 。对Runge-Kutta格式作这一修改的后果是:只有当 $\alpha_{m-1} = 1/2$ 时才能实现二阶时间精度;否则ï¼多步格式在时间上只有一阶精度。en

Unlike in the classical Runge-Kutta schemes, only the zeroth solution and the newest residual are stored here in order to reduce the memory requirements. The stage coefficients can be tuned to increase the maximum time step and to improve the stability for a particular spatial discretisation [2]-[4]. For consistency, it is only required that $\alpha_m = 1$ . A consequence of the modification to the Runge-Kutta scheme is that second-order time accuracy can be realised only if $\alpha_{m-1} = 1/2$ . Otherwise, the multistage scheme is first-order accurate in time.

上述多步方法(6.5)特别适合于结构网格和非结构网格上的上风空间离散化。中心离散化格式与下面要介绍的混合多步方法配合使用则效率更高。表6.1给出一阶和二阶上风格式在3步至5步格式下的优化阶段系数[2]。实践经验表明ï¼当流场中含有强激波时ï¼无论空间离散化的阶数如何ï¼都应优先选用一阶格式的系数。这可以解释为:任何高阶格式都会在激波处降为一阶ï¼以防止解的振荡;而强激波处的残差对收敛到定态的影响最为显著。en

The above multistage approach (6.5) is particularly suitable for upwind spatial discretisation on structured as well as unstructured grids. Central discretisation schemes perform more efficiently with the hybrid multistage methodology, which will be described next. Sets of optimised stage coefficients for first- and second-order upwind schemes are presented in Table 6.1 for three- to five-stage schemes [2]. Practical experience shows that the coefficients for the first-order scheme should be preferred in cases, where the flow field contains strong shocks, regardless of the order of the spatial discretisation. This can be explained by the fact that every higher-order scheme switches to first order at shocks to prevent oscillations of the solution. However, the residuals at strong shocks influence the convergence to steady state most significantly.

(见原书第185页表6.1ï¼多步格式:一阶与二阶上风空间离散化的优化阶段系数(α)与CFL数(σ))

          first-order scheme       second-order scheme
stages  3      4      5       3      4      5
sigma   1.5    2.0    2.5     0.69   0.92   1.15
alpha_1 0.1481 0.0833 0.0533  0.1918 0.1084 0.0695
alpha_2 0.4000 0.2069 0.1263  0.4929 0.2602 0.1602
alpha_3 1.0000 0.4265 0.2375  1.0000 0.5052 0.2898
alpha_4        1.0000 0.4414         1.0000 0.5060
alpha_5               1.0000                1.0000

表6.1:多步格式:一阶与二阶上风空间离散化的优化阶段系数(α)与CFL数(σ)。

(见原书第185页表6.2ï¼混合多步格式:中心格式与上风空间离散化的优化阶段系数(α)、混合系数(β)及CFL数(σ))

         central scheme    1st-order upwind   2nd-order upwind
         sigma = 3.6       sigma = 2.0        sigma = 1.0
stage    alpha   beta      alpha   beta       alpha   beta
1        0.2500  1.00      0.2742  1.00       0.2742  1.00
2        0.1667  0.00      0.2067  0.00       0.2067  0.00
3        0.3750  0.56      0.5020  0.56       0.5020  0.56
4        0.5000  0.00      0.5142  0.00       0.5142  0.00
5        1.0000  0.44      1.0000  0.44       1.0000  0.44

表6.2:混合多步格式:中心格式与上风空间离散化的优化阶段系数(α)、混合系数(β)及CFL数(σ)。注意:一阶与二阶上风格式的系数完全相同ï¼但CFL数不同。

一切显式格式的主要缺点是:时间步长( $\Delta t$ )受到控制方程的特征以及网格几何的严格限制。我们将在6.1.4小节讨论最大允许时间步长的计算。时间步长和所谓CFL数(CFL number)确定的理论问题将在10.3节稳定性分析中考察。en

The main disadvantage of every explicit scheme is that the time step ( $\Delta t$ ) is severely restricted by the characteristics of the governing equations as well as by the grid geometry. We shall discuss the computation of the maximum allowable time step in Subsection 6.1.4. Theoretical aspects of the determination of the time step and the so-called CFL number will be considered in Section 10.3 on stability analysis.