Spatial modeling issues realizability

In the remainder of this section, we introduce the principal modeling issues related to spatial transport using moment methods. First, we discuss the realizability of the NDF and of moment sets (which are related to the numerical errors discussed above). Second, we introduce the phenomenon of particle trajectory crossing (PTC) that occurs with the inhomogeneous KE (and is exactly captured by the NDF), and describe how it leads to a closure problem in the moment-transport equations. Next, we look at issues related to coupling between spatial and phase-space transport in the GPBE (i.e. due to correlations between velocity and internal coordinates such as particle volume). Finally, we introduce KBFVM for solving the moment-transport equations in connection with QBMM, and briefly discuss how they can be used to ensure realizability as well as to capture PTC and to treat coupled moments. 

The alternative to the hierarchical modeling is known as concurrent modeling and consists of combining different numerical models that simultaneously describe different sub-domains. Each numerical model runs in its sub-domain and exchanges information with its parent/child sub-domain at defined boundaries. The critical issue here is to define the criteria and protocols in order to automate the application of more detailed numerical models in a simulation domain in time and realizing the so-called temporal multi-scale calculations or/ and space referred as spatial multi-scale ones. 

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