4.3.5 Limiter Functions 限制器函数[cfd-4-3-5]

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 are looking for 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. Or in other words, if the initial data is monotone then the solution has to remain monotone. The rather stringent conditions for monotonicity preserving schemes (or the more rigorous ones for TVD schemes) are often given up in favour of the Local Extremum Diminishing (LED) conditions [60]. Here, a local extremum contained only within the stencil has to decrease.

However, due to Godunov's theorem there is no possibility for a higher-order linear scheme (such as the MUSCL approach) to be monotonicity preserving [90]. It is therefore necessary to employ non-linear limiter functions in order to construct a monotonicity preserving or a TVD discretisation. This is demonstrated in Fig. 4.10, where the upwind TVD scheme of Eq. (4.98) was used with and without a limiter to compute 2-D transonic flow past the NACA 0012 airfoil. It can be clearly seen that without limiter, the solution exhibits large oscillations in the neighbourhood of the shocks on the upper and the lower side of the airfoil. On the other hand, the limited and the unlimited solutions become nearly identical away from the shocks.

The purpose of a limiter is to reduce the slopes (i.e., \((U_{I+1} - U_I)/\Delta x\)) used to interpolate a flow variable to the face of a control volume in order to constrain the solution variations. At strong discontinuities, the limiter has to reduce the slopes to zero to prevent the generation of new extrema. This implies for the MUSCL approach as well as for the TVD schemes that the (monotone) first-order upwind scheme (\(\epsilon = 0\) in Eq. (4.46)) is recovered in the immediate vicinity of large gradients. The last requirement to be imposed on a limiter is quite obvious - the original unlimited discretisation has to be obtained in smooth flow regions, in order to keep the amount of numerical dissipation as low as possible. The effect of a limiter on the interpolation of the left and right states is sketched in Fig. 4.11. The example shows the slope reduction at the local minimum at \(I\) and the change of the slope at the cells \((I+1)\), \((I+2)\) to achieve a monotone solution. It is important to realise that a difference between the left and right state at a face may (and generally will) still be present.

In the following, we shall describe four different limiter functions, which are well-established and proven in practice. We shall consider limiters for the second-order MUSCL, for the CUSP and for the upwind TVD scheme.

Van Leer's MUSCL approach [29] is turned into a monotonicity preserving scheme by employing a limiter function to reduce the differences \(\Delta_{+}U_I\) and \(\Delta_{-}U_I\) in Eq. (4.47) when necessary. Introducing slope limiters \(\Phi^{\pm}\), the MUSCL interpolation formulae in Eq. (4.46) are modified as follows (see also Fig. 4.8)

The parameter \(\epsilon\) in Eq. (4.46) was set equal to unity. The slope limiters are functions of the ratios of the consecutive solution variations, i.e., \(\Phi^{\pm}_{I+1/2} = \Phi(r^{\pm}_{I+1/2})\), with the definitions [1]

If we substitute now \(r_L\) for \(r^{+}_{I-1/2}\) and \(r_R\) for \(r^{-}_{I+3/2}\), thus

we can write Eq. (4.104) in the form

The above relationships Eq. (4.107) can be simplified if we consider only slope limiters with the symmetry property

With this definition, the limited MUSCL interpolation Eq. (4.104) becomes [91]

with the limiter function defined as

Different formulations of the slope limiter \(\Phi\) in Eq. (4.110) are now possible, which can be tailored to specific values of \(\hat{\kappa}\) to give the most accurate but stable and monotonicity preserving MUSCL scheme.

One particularly suitable combination for the second-order, upwind-biased scheme with \(\hat{\kappa} = 0\) is [92]

In this case, the function \(\Psi(r)\) corresponds to the Van Albada limiter [93]

and we obtain with Eq. (4.109) the following expressions for the left and right state

The function \(\delta\) is formally identical for both states. It reads

The coefficients \(a\) and \(b\) are defined for the left and right state as

and the difference operators \(\Delta_{\pm}\) are given by Eq. (4.47). The additional parameter \(\epsilon\) in Eq. (4.114) prevents the activation of the limiter in smooth flow regions due to small-scale oscillations [92]. This is sometimes necessary in order to achieve a fully converged steady-state solution. The parameter \(\epsilon\) is conveniently set proportional to the local grid scale, in 3D for example to \(\Omega^{1/3}\) [92], [94]. Additional scaling of the parameter \(\epsilon\) is required if the particular state variable \(U\) is given in physical units. It can be shown that the relations in Eq. (4.113) are identical to the original (unlimited) MUSCL scheme (4.46) with \(\hat{\kappa} = 0\) for smoothly varying flow. Thus, the accuracy of the solution is not influenced. On the other hand, the function \(\delta\) becomes zero at local extrema, reducing the accuracy to first order as desired.

Another limiter function was devised for the three-point, second-order accurate upwind-biased MUSCL scheme with \(\hat{\kappa} = 1/3\). Here, the slope limiter is given by

In this case, the function \(\Psi(r)\) corresponds to the limiter of Hemker and Koren [95]. Following the same way as in the previous case, we obtain formulae for the left and right state at the face \((I+1/2)\) which are identical to Eq. (4.113), but now with [92]

The definitions of the coefficients \(a\), \(b\), and of the parameter \(\epsilon\) are retained.

In the framework of the CUSP scheme (Subsection 4.3.2), the left (\(L\)) and right (\(R\)) states are evaluated to second-order accuracy according to [43]

where

In the above Eqs. (4.118) and (4.119), \(U\) represents a dependent variable and \(L()\) the limited average

respectively. The limiter itself is defined as

where \(\sigma\) is a positive coefficient which is usually set equal to two. The constant \(\epsilon\) is required to prevent division by zero (e.g., \(\epsilon = 10^{-20}\)). If \(\Delta_1\) and \(\Delta_2\) happen to have opposite sign but the same magnitude, the limiter becomes \(\Psi = 0\). This means that we obtain only a first-order accurate approximation for the left and the right state.

It should be mentioned that it is also possible to employ the MUSCL scheme with \(\hat{\kappa} = 0\) and the Van Albada limiter from Eqs. (4.113)-(4.115) instead of the above relations.

In comparison to the previous cases, the limiter here acts not on the conservative or the primitive variables, but on the characteristic variables \(\vec{C}\). One particularly suitable limiter function is given by [84]

where the \(\Delta C^{l}_{I+1/2}\) represents the difference of the characteristic variables at face \((I+1/2)\) of the control volume (Eq. (4.101)). The positive constant \(\epsilon \approx 10^{-20}\) in the denominator prevents division by zero. In regions with high gradients, the limiter function becomes zero, which leads with Eq. (4.99) and Eq. (4.98) to first-order accurate upwind scheme. The upwind TVD scheme in Eq. (4.98) retains second-order accuracy in areas with smoothly varying flow variables, where \(\Psi^{l}_{I} = C^{l}_{I} - C^{l}_{I-1}\).