Upwind differencing scheme first order

In addition to the central differencing and upwind differencing schemes, which are first-order schemes, another popular finite difference scheme is the QUICK scheme, a second-order upwind differencing scheme. Higher order means that more node points are involved when estimating the values of the dependent variables and their derivatives for formulating the finite difference equations. 

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. 

With standard DQMOM, when the explicit Euler scheme in time and the first-order upwind differencing scheme for space are employed, the volume-average weights in the cell centered at X at time (n + l)Af are... 

If a first-order upwind spatial-differencing scheme is used, the reconstructed moments tW/t,i i/2 at the cell interfaces are equal to the upwind values and, in the case of positive u, the following equation is obtained ... 

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