The collision term for integer moments

For quadrature-based moment methods, the collision terms for integer moments are needed. For this special case, the method developed in Fox Vedula (2010) leads to closed expressions for the integrals in terms of finite sums. For integer moments, we have il/ y) = Vj 2 Vg and the terms in the sums over n in Eqs. (6.55)-(6.57) are zero for n max(/i, /2. h)- For this particular case, we will define for m = 0 by 

Using the definitions of the operators in Section 6.1.4, we can rewrite these terms as polynomials in the components of vi  

The reader will recognize that the coefficients are related to the integrals over the collision angles discussed in Section 6.1.4. The exact definitions are 

These coefficients, and the summations in Eq. (6.60), are most easily computed using a symbolic math program. Some examples of co, v, g) for selected moments up to fifth order are given in Tables 6.1-6.9. For clarity, in these tables we have denoted the velocity difference vector by g = gi,g2,g3) and g =g + gl+gy The final expressions for the collision source terms for integer moments of order j = h+h + h can now be written in the form of Eq. (6.54) as 

The reader can verify that the source terms for the zeroth- and first-order moments are exactly zero (i.e. the conservation of mass and mean momentum). It is interesting to 

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