Realizability advection scheme

When the first-order scheme is semi-implicit with 0 < a < 1, this realizability condition is a mix of those found for pure advection and pure diffusion. 

The first class is a generalization of the advection-diffusion problem discussed in Section 8.3, and much of the material developed there can be reformulated to develop a realizable scheme for the GPBE. The second class is a generalization of the kinetic equation considered in Section 8.4, and has been referred to in the literature as the semi-kinetic model (Laurent et al, 2004 Laurent Massot, 2001). In the following, we will treat each class separately, although the reader will undoubtedly note many similarities between the numerical schemes. A third class of GPBE, lying between the two listed above, can also be identified wherein the scalar-dependent convection velocity has a parametric form such as... 

Since the moment-transport equation is closed, one might be tempted to try to design a high-order scheme directly for M. However, for such a scheme it would be difficult to ensure realizability, and thus a better approach is to work directly with the NDF. Moreover, when the advection velocity depends on f the moment-transport equation is not closed, thus working with the NDF will result in more general formulas. 

In summary, in this section we have shown that the realizable scheme for mixed advection is very similar to those introduced for advection and free transport. However, the numerical properties of the scheme are very dependent on the functional form used for A. In order for the standard models for mixed advection to be well conditioned, the functional form for A must compensate for velocity abscissas located too close to the origin. With multivariate FQMOM, the model given in Fq. (B.56) can be used for this purpose. The potential for singular behavior with mixed advection makes it nevertheless problematic, and alternative schemes may be required to properly handle such behavior. 

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