3.2 Temporal Discretisation 时间离散[cfd-3-2]

A basic explicit time-integration scheme is obtained by setting \(\beta=0\) and \(\omega=0\) in Eq. (3.4). In this case, the time derivative is approximated by a forward difference and the residual is evaluated at the current time level only (based on known flow quantities), i.e.,

where the mass matrix was lumped. This represents a single-stage scheme, because a new solution \(\vec{W}^{n+1}\) results from only one evaluation of the residual. The scheme Eq. (3.7) is of no practical value, since it is stable only if combined with a first-order upwind spatial discretisation.

Very popular are multistage time-stepping schemes (Runge-Kutta schemes), where the solution is advanced in several stages [64] and the residual is evaluated at intermediate states. Coefficients are used to weight the residual at each stage. The coefficients can be optimised in order to expand the stability region and to improve the damping properties of the scheme and hence its convergence and robustness [64], [112], [113]. Also, depending on the stage coefficients and the number of stages, a multistage scheme can be extended to 2nd- or higher-order accuracy in time. Special Runge-Kutta schemes were also designed to preserve the properties of the TVD and ENO spatial discretisation methods, while maximising the allowable time step [114].

Explicit multistage time-stepping schemes can be employed in connection with any spatial discretisation scheme. They can be easily implemented on serial, vector, as well as on parallel computers. Explicit schemes are numerically cheap, and they require only a small amount of computer memory. On the other hand, the maximum permissible time step is severely restricted because of stability limitations. Particularly for viscous flows and highly stretched grid cells, the convergence to steady state slows down considerably. Furthermore, in the case of stiff equation systems (e.g., real gas simulation, turbulence models), or of stiff source terms, it can take extremely long to achieve the steady state. Or even worse, an explicit scheme may become unstable or lead to spurious steady solutions [115].

If we are interested in steady-state solutions only, we can select from (or combine) several convergence acceleration methodologies. The first, and very common, technique is local time-stepping. The idea is to advance the solution in each control volume with the maximum allowable time step. As a result, the convergence to the steady state is considerably accelerated. However, the transient solutions are no longer temporally accurate. Another approach is the so-called characteristic time-stepping. Here, not only locally varying time steps are used, but also each equation (continuity, momentum and energy equation) is integrated with its own time step. The potential of this concept was presented for 2-D Euler equations in Ref. [116]. A further acceleration technique, which is similar to the characteristic time-stepping is Jacobi preconditioning [117]-[119]. It is basically a point-implicit Jacobi relaxation, which is carried out at each stage of a Runge-Kutta scheme. Jacobi preconditioning can be seen as a time-stepping in which all wave components (eigenvalues of the flux Jacobian) are scaled to have the same effective speed. It also adds an implicit component to the basic explicit scheme.

Another very popular acceleration method is aimed at increasing the maximum possible time step by introducing a certain amount of implicitness in the explicit scheme. It is termed implicit residual smoothing or residual averaging [120], [121]. On a structured grid, the method requires the solution of a tridiagonal matrix for each conservative variable. In the case of unstructured grids, the matrix is usually inverted by means of Jacobi iteration. The standard implicit residual smoothing allows an increase of the time step by a factor of 2-3. Several other implicit residual smoothing techniques were developed. For example the upwind implicit residual smoothing methodology [122], which was designed to be employed together with an upwind spatial discretisation. In comparison to the standard technique, it allows for significantly larger time steps and it also improves the robustness of the time-stepping process [123]. One further method is the implicit-explicit residual smoothing [124], [125], which is intended to improve the damping properties of the time discretisation at larger time steps.

The last and probably the most important convergence acceleration technique, which should be mentioned here, is the multigrid method. It was developed in the 1960's in Russia by Fedorenko [126] and Bakhvalov [127]. They applied multigrid for the solution of elliptic boundary-value problems. The methodology was further advanced and promoted by Brandt [128], [129]. The idea of multigrid is based on the observation that iterative schemes usually eliminate high-frequency errors in the solution (i.e., oscillations between the control volumes) very effectively. On the other hand, they perform quite poor in reducing low-frequency (i.e., global) solution errors. Therefore, after advancing the solution on a given grid, it is transferred to a coarser grid, where the low-frequency errors become partly high-frequency ones and where they are again effectively damped by an iterative solver. The procedure is repeated recursively on a sequence of progressively coarser grids, where each multigrid level helps to annihilate a certain bandwidth of error frequencies. After the coarsest grid is reached, the solution corrections are successively collected and interpolated back to the initial fine grid, where the solution is then updated. This complete multigrid cycle is repeated until the solution changes less than a given threshold. In order to accelerate the convergence even further, it is possible to start the multigrid process on a coarse grid, carry out a number of cycles and then to transfer the solution to a finer grid, where the multigrid cycles are performed again. The procedure is then successively repeated until the finest grid is reached. This methodology is known as Full Multigrid (FMG) [129].

As already mentioned, the multigrid method was originally developed for the solution of elliptic boundary-value problems (Poisson equation), where it is very efficient. Jameson first proposed to employ multigrid also for the solution of the Euler equations [120], [130]. The approach was based on the so-called Full Approximation Storage (FAS) scheme [129], where multigrid is directly applied to the non-linear governing equations. Nowadays, multigrid represents a standard acceleration technique for the solution of the Navier-Stokes equations. Examples of implementations can be found in Refs. [131]-[136] for structured grids, and in Refs. [137]-[146] for unstructured grids. Although not as fast as in the case of elliptic differential equations, it was often demonstrated that multigrid can accelerate the solution of the Euler or the Navier-Stokes equations by a factor between 5 and 10. An example for transonic flow is shown in Fig. 3.7. Recent research also revealed that faster convergence can be achieved if the governing equations are decomposed into hyperbolic and elliptic parts [147]. We shall return to the multigrid methodology again in Section 9.4.