Application of quadrature to collision terms

The moment-transport equations discussed above become more and more complicated as the order increases. Moreover, these equations are not closed. In quadrature-based moment methods, the velocity-distribution function is reconstructed from a finite set of moments, thereby providing a closure. In this section, we illustrate how the closure hypothesis is applied to solve the moment-transport equations with hard-sphere collisions. For clarity, we will consider the monodisperse case governed by Eq. (6.131). Formally, we can re-express this equation in conservative form  

The quadrature-based closure of Eqs. (6.176) and (6.177) then proceeds as follows. Let / (f, X, v) denote the velocity-distribution function reconstructed from the transported moments (Eq. 6.173), and define the negative- and positive-integer moments for 

Analogous moments can be defined for directions X2 and X3. Since / is a known function, these moments can be computed (usually analytically) for any choice of (h,h,h)- Thus, using each set of moments, we can construct an arbitrarily high-order quadrature representation that can be used to evaluate the integrals in Eqs. (6.176) and (6.177). Formally, we can express the reconstruction procedure as 

An important point that is evident from the expressions in Eqs. (6.179) and (6.180) is that it suffices to know the functional form of / in order to account for the effect of 

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