6.1.4 Determination of the Maximum Time Step 最大时间步长的确定[cfd-6-1-4]

Every explicit time-stepping scheme remains stable only up to a certain value of the time step \(\Delta t\). To be stable, a time-stepping scheme has to fulfil the so-called Courant-Friedrichs-Lewy (CFL) condition [19]. It states that the domain of dependence of the numerical method has to include the domain of dependence of the partial differential equation. The CFL condition means for the basic explicit scheme (6.4) that the time step should be equal to or smaller than the time required to transport information across the stencil of the spatial discretisation scheme. Hence, in 1D the condition for the time step would read for the linear convection equation

where \(\Delta x/|\Lambda_c|\) represents the time necessary to propagate information over the cell size \(\Delta x\) with the velocity \(\Lambda_c\). The velocity \(\Lambda_c\) corresponds to the maximum eigenvalue of the convective flux Jacobian. The positive coefficient \(\sigma\) denotes the CFL number. The magnitude of the CFL number depends on the type and the parameters of the time-stepping scheme, as well as on the form of the spatial discretisation scheme. We shall investigate the dependency of \(\sigma\) in Section 10.3 for two model problems. Tables 6.1 and 6.2 list the CFL numbers for various multistage schemes and discretisations.

The maximum time step can be determined for linear model equations with the aid of Von Neumann stability analysis (Section 10.3). However, the maximum time step can be calculated only approximately in multiple dimensions and for non-linear governing equations. In the following, we will present relations for the estimation of the time step on structured and unstructured grids, and for inviscid as well as for viscous flows.

On a structured grid, the time step \(\Delta t\) can be determined for a control volume \(\Omega_I\) from the approximate relation [20]-[22]

The CFL number \(\sigma\) is given for multistage schemes in Table 6.1 and for hybrid schemes in Table 6.2. The spectral radii of the convective flux Jacobians (A.9) read for the three grid directions

The normal vectors and face areas in Eq. (6.15) are obtained by averaging the corresponding values from the two opposite sides of the control volume in the respective direction. For example, if a dual control volume (Subsection 4.2.3) would be oriented as sketched in Fig. 4.1b, we would use in the \(I\)-direction

Similar expressions hold for the \(J\)- and \(K\)-direction. With Eq. (6.14), we obtain a local time step, which is valid for one control volume only. If we are interested in a steady state solution, we may use the local time step to accelerate the convergence (cf. Section 9.1). However, if time accuracy is important, we have to employ one global time step for all volumes, i.e.,

where the minimum over all control volumes is taken.

For viscous flows, the spectral radii of the viscous flux Jacobians (A.10) have to be included in the computation of \(\Delta t\). They can severely limit the maximum time step in boundary layers. The time step can be evaluated from [5], [21], [22]

The constant which multiplies the viscous spectral radii is usually set as \(C = 4\) for central spatial discretisations, \(C = 2\) for first-order upwind and \(C = 1\) for second-order upwind discretisations. If we assume that an eddy-viscosity turbulence model is employed, the viscous spectral radii are given by [21], [22]

and similarly for the other directions. In Equation (6.19), \(\mu_L\) denotes the laminar and \(\mu_T\) the turbulent dynamic viscosity coefficient, respectively. Furthermore, \(Pr_L\) and \(Pr_T\) are the laminar and the turbulent Prandtl numbers. The CFL numbers in Tables 6.1 and 6.2 apply also for viscous flows. Particularly efficient for viscous flows is the (5,3) hybrid scheme from Eq. (6.7).

Several approaches were suggested for the estimation of the maximum time step on unstructured grids. We shall present two different approaches below.

One proven method, which closely follows the implementation on structured grids, reads [6]

where \(\hat{\Lambda}_c\) and \(\hat{\Lambda}_v\) represent a sum of the convective and viscous spectral radii over all faces of the control volume. As on structured grids, \(1 \le C \le 4\) is usually used. In the case of the cell-centred scheme, the spectral radii are defined as [6]

The values of the flow variables at the faces of the control volume are obtained by arithmetic averaging.

The spectral radii predicted by Eq. (6.21) are too large, particularly on mixed element grids. This leads to a smaller time step than required for the stability. The implementation in Ref. [23] offers a more accurate estimation of the time step, namely

with the convective spectral radii

and with the viscous spectral radii (eddy-viscosity turbulence model assumed)

The variables \(\Delta\hat{S}^x\), \(\Delta\hat{S}^y\) and \(\Delta\hat{S}^z\), respectively, represent projections of the control volume on the \(y\)-\(z\)-, \(x\)-\(z\)- and the \(x\)-\(y\)-plane. They are given by the formulae

where \(S_x\), \(S_y\) and \(S_z\) denote the \(x\)-, \(y\)- and the \(z\)-component of the face vector \(\vec{S} = \vec{n}\cdot\Delta S\). The CFL numbers stay in general the same as on structured grids. Thus, the values collected in Tables 6.1 and 6.2 still apply. Furthermore, the convergence to steady state can also be accelerated by local time stepping, in the same way as on the structured grids. A global time step, necessary for simulating unsteady flows, can be obtained from Eq. (6.17) as before.