Standard finite-volume schemes for moments

In this section, we develop a high-order scheme to solve Eq. (8.63) on a uniform mesh with spacing Ax. Denoting the mesh points by x , the volume-average moments are 

Using a first-order upwind reconstruction for the advection term in Eq. (8.65), we have 

Using central differences, the gradient terms are approximated by 

This system of coupled equations must be solved by integrating forward in time starting from the initial conditions mk,i(0). Eor stability, the time integration of the diffusion term can be treated implicitly using, for example, the Crank-Nicolson (CN) scheme. If we denote the volume-average moments at time t = n At by , a semi-implicit scheme for 

Writing Eq. (8.69) as = Bm , the formal solution for the updated moments is thus 

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