Realizable finite-volume schemes for moments

In this section, we discuss realizable schemes for Eq. (8.61) that are based on the KBFVM presented in Section 8.2 and Appendix B. In order to implement these methods, we must reconstruct the NDE from the moments. For this purpose, we will employ gamma EQMOM (Eq. (3.84)) with N = 2 nodes, with which the reconstructed moments are given by 

The Gauss-Laguerre quadrature is defined such that Wma- o ap = a- Furthermore, the second summation in Eq. (8.74) corresponds to the moments of the gamma PDF, 

The first-order scheme for convection is exactly the same as the one described in Section 8.3.2. The only difference (which is not strictly necessary) is that we will replace the 

The constant 0 in the Stokes-Einstein diffusivity is added to treat the limit of vanishing size so that the diffusivity coefficient remains finite. The realizability condition for Eq. (8.75) is 

Note that it is not possible to write an implicit form for the diffusion term with Stokes-Einstein diffusivity because the weights and abscissas at time n + I depend in a highly nonlinear manner on Also, we should note that a more accurate representation of the Stokes-Einstein (or any -dependent) diffusion coefficient can be attained by increasing M. The reader should keep in mind that increasing M does not change the number of moments that need to be solved for, and thus the computational cost is relatively insensihve to the value of M. 

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