Finite-volume scheme conservative

In this section the numerical conservation properties of finite volume schemes for inviscid incompressible flow are examined. Emphasis is placed on the theory of kinetic energy conservation. Numerical issues associated with the use of kinetic energy non-conservative schemes are discussed [158, 49, 47]. 

The finite-volume representation of the mixed moments in each cell is found by computing the volume average of Eq. (8.110). On the right-hand side die volume averages of products such as mo o.o appear. Thus, if Mo is piecewise constant in the cell, the volume-average product depends on the volume-average moments, which are the conserved quantities in the finite-volume scheme. 

In these equations fi is the coluirm mass of dry air, V is the velocity (u, v, w), and (jf) is a scalar mixing ratio. These equations are discretized in a finite volume formulation, and as a result the model exactly (to machine roundoff) conserves mass and scalar mass. The discrete model transport is also consistent (the discrete scalar conservation equation collapses to the mass conservation equation when = 1) and preserves tracer correlations (c.f. Lin and Rood (1996)). The ARW model uses a spatially 5th order evaluation of the horizontal flux divergence (advection) in the scalar conservation equation and a 3rd order evaluation of the vertical flux divergence coupled with the 3rd order Runge-Kutta time integration scheme. The time integration scheme and the advection scheme is described in Wicker and Skamarock (2002). Skamarock et al. (2005) also modified the advection to allow for positive definite transport. 

The finite volume method has become a very popular method of deriving discretizations of partial differential equations because these schemes preserve the conservation properties of the differential equation better than the schemes based on the finite difference method. 

An alternative to the above modeling approach is to simulate thermal radiation exchange using a conservative variant of the discrete ordinates (DO) radiation model, called the finite-volume (FV) scheme, implemented in the Fluent software package. 

Filbet and Laurengot [58] developed a particular finite volume method (FVM) scheme for dicretizing the Smoluchowski equation for purely coalescing systems. For the application of the FVM they established a continuous flux form of the PBE coalescence source terms. The FVM thus ensures that the poly-disperse particle fluxes between the individual sections are conserved in the system. This approach deviates from the conventional sectional methods which are applied to the standard discrete form of the PBE source terms. Kumar et al. [113] adapted the FVM scheme for solving the transformed coalescence source terms to pure breakage and simultaneous breakage and coalescence systems. 

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