Finite-volume scheme first order

To illustrate the principles of the finite volume method, as a first approach, the implicit upwind differencing scheme is used for a multi-dimensional problem. Although the upwind differencing scheme is very diffusive, this scheme is frequently recommended on the grounds of its stability as the preferred method for treatment of convection terms in multiphase flow and determines the basis for the implementation of many higher order upwinding schemes. 

The thermodynamics of the I-N phase transition has been extensively investigated for resolving the issue concerning the order of the transition. Following the Ehrenfest scheme, a phase transition is classified into a first-order transition or a second-order one, depending upon the observation of finite discontinuities in the first or the second derivatives of the relevant thermodynamic potential at the transition point. An experimental assessment of the order of the I-N transition has turned out to be not a simple task because of the presence of only small discontinuities in enthalpy and specific volume. It follows from high-resolution measurements that I-N transition is weakly first order in nature [85]. 

Finite Volume Methods The finite volume method, when the permeabihty tensor is diagonal in the selected coordinate system, approximates the pressure and saturation functions as piecewise constant in each grid block. The flux components are assumed constant in their related half-cells. Thus when two cells are joined by a face, the related component of flux is assumed to be the same each side of the face. The balance laws are invoked separately on each grid block, and are discretised in time either by an explicit or fully implicit first order Euler scheme or other variant as discussed in the previous subsection. 

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