5.3.5 Limiter Functions 限制器函数[cfd-5-3-5]

Second- and higher-order upwind spatial discretisations require the use of so-called limiters or limiter functions in order to prevent the generation of oscillations and spurious solutions in regions with large gradients (e.g., at shocks). Hence, what we want to achieve is at least a monotonicity preserving scheme. This means that maxima in the flow field must be non-increasing, minima non-decreasing, and no new local extrema may be created during the time evolution. We discussed this point in Subsection 4.3.5 for the case of structured upwind discretisation schemes.

On unstructured grids, the purpose of a limiter is to reduce the gradients used to reconstruct the left and right state at the face of the control volume. The limiter function must be zero at strong discontinuities, in order to obtain a first-order upwind scheme which guarantees monotonicity. Setting the limiter to zero leads to the constant reconstruction of Eq. (5.38). Of course, the original unlimited reconstruction has to be retained in smooth flow regions, in order to keep the amount of numerical dissipation as low as possible. In the following, we shall describe two widely used limiter functions - namely the limiters of Barth and Jespersen [30], and of Venkatakrishnan [68], [69].

The first implementation of a limiter function on unstructured grids was presented in Ref. [30]. In the case of the median-dual scheme, it is defined at the node \(i\) as

with the abbreviations

In Equations (5.64) and (5.65), \(\min_j\) or \(\max_j\) means the minimum or maximum value of all direct neighbours \(j\) of node \(i\) (i.e., all nodes connected to \(i\) by an edge). Furthermore, the edge vector \(\vec{r}_{ij}\), which is shown in Fig. 5.9 or in Fig. 5.13b, is defined according to Eq. (5.43). Finally, \(U_j\) denotes a scalar quantity at some neighbouring node \(j\). Similar formulae to those above hold for the cell-centred scheme with cell instead of node indices and with

where \(\vec{r}_L\) denotes the vector from the cell-centroid to the midpoint of the corresponding cell face. In order to avoid division by a very small value of \(\Delta_2\) in Eq. (5.64), it is better to modify \(\Delta_2\) as \(\mathrm{Sign}(\Delta_2)(\left|\Delta_2\right| + \omega)\), where \(\omega\) is approximately the machine accuracy [68].

Barth's limiter enforces a monotone solution. However, it is rather dissipative and it tends to smear discontinuities. A further problem presents the activation of the limiter due to numerical noise in smooth flow regions. This usually prevents the full convergence to steady state [68], [38]. Therefore, the limiter function due to Venkatakrishnan became more popular.

Venkatakrishnan's limiter [68], [69] is widely used because of its superior convergence properties. The limiter reduces the reconstructed gradient \(\nabla U\) at the vertex \(i\) by the factor

where

In the above Eq. (5.68), \(U_{max}\) and \(U_{min}\) stand for the minimum/maximum values of all surrounding nodes \(j\) and including the node \(i\) itself. Definitions of \(U_{max}\), \(U_{min}\) and \(\Delta_2\) are given in Eq. (5.65). The parameter \(\epsilon^{2}\) is intended to control the amount of limiting. Setting \(\epsilon^{2}\) to zero results in full limiting, but this may stall the convergence. Contrary to that, if \(\epsilon^{2}\) is set to a large value, the limiter function will return a value of about unity. Hence, there will be no limiting at all and wiggles could occur in the solution. In practice, it was found that \(\epsilon^{2}\) should be proportional to a local length scale, i.e.,

where \(K\) is a constant of \(\mathcal{O}(1)\) and \(\Delta h\) is for example the cube-root of the volume (square-root of the area in 2D) of the control volume. It is important to notice that the limiter function (5.67) must be defined with non-dimensional quantities. The influence of the coefficient \(K\) in Eq. (5.69) on the resolution of a shock is demonstrated in Fig. 5.16. It can be seen that the fully limited \((K = 0)\) and the solution for \(K = 5\) are identical. However, the explicit time-stepping scheme converged only about three orders of magnitude for \(K = 0\), whereas for \(K = 5\) it converged to machine zero (Fig. 5.17). Figure 5.16 also shows that the solution becomes gradually unlimited with increasing values of \(K\). This manifests itself as an increasing overshoot at the shock.

The computational effort for the evaluation of one of the above limiter functions is relatively high. Two loops over edges (faces in the case of the cell-centred scheme) and one loop over nodes (cells) are necessary in order to compute \(U_{max}\), \(U_{min}\) as well as the limiter \(\Psi\) itself. Furthermore, \(U_{max}\), \(U_{min}\) and \(\Psi\) have to be stored node-(cell-)wise, separately for each flow variable.