Upwinding formula

In a similar fashion, more accurate higher order formulas can be developed. The four point upwind formula is... 

Different combinations of spatial finite difference formulas were tried to determine the best set for our system of equations. The two point upwind formula was found to be best for the solids component molar fluxes. The low order formula was used because most of the gasifier reactions turn off abruptly when a component disappears and this creates sharp discontinuities. Higher order formulas tend to flatten out discontinuities, and in some cases, this causes material balances to be lost which then leads to numerical instability problems. Maintaining component material balance is an important aid to preserving numerical stability in the calculations. The low order formulas minimized these difficulties. 

The four point upwind biased formula worked best for the solids stream energy flux calculation. Some downstream information was useful because of the countercurrent flow of the gas and solids streams. To keep the same order, the four point downwind biased formula was used at the top of the reactor and the four point upwind formula was used at the bottom. 

The Lagrange interpolation polynomial was again used to develop the finite difference formulas. To avoid additional iterations, only upwind differences were used. The two point upwind formula for the solids stream concentration variable at any location z within the reactor for time t is given by... 

The same 82 point variable grid structure was used in the time method of lines calculations as was used for the distance method of lines calculations. Also, the three and four point upwind formulas were found to attenuate the calculated step responses too much and they were discarded. 

Virtual sources As indicated above, the gaussian model was formulated for an idealized point source, and such an approach may be unnecessarily conservative (predict an unrealistically large concentration) for a real release. There are formulations for area sources, but such models are more cumbersome than the point source models above. For point source models, methods using a virtual source have been proposed in the past which essentially use the maximum concentration of the real source to determine the location of an equivalent upwind point source that would give the same maximum concentration at the real source. Such an approach will tend to overcompensate and unrealistically reduce the predicted concentration because a real source has lateral and along-wind extent (not a maximum concentration at a point). Consequently, the modeled concentration can be assumed to be bounded above, using the point source formulas in Eq. (23-78) or (23-79), and bounded below by concentrations predicted by using a virtual source approach. 

The convective terms are approximated using an upwind difference formula. For example,... 

First order hyperbolic differential equations transmit discontinuities without dispersion or dissipation. Unfortunately, as Carver (10) and Carver and Hinds (11) point out, the use of spatial finite difference formulas introduces unwanted dispersion and spurious oscillation problems into the numerical solution of the differential equations. They suggest the use of upwind difference formulas as a way to diminish the oscillation problem. This follows directly from the concept of domain of influence. For hyperbolic systems, the domain of influence of a given variable is downstream from the point of reference, and therefore, a natural consequence is to use upstream difference formulas to estimate downstream conditions. When necessary, the unwanted dispersion problem can be reduced by using low order upwind difference formulas. 

The four point upwind biased formula is given by... 

This restriction allowed the coefficients for the four point upwind biased formula to sequence through the following values for one grid spacing change... 

The functional relationship of the HRIC scheme is also function of the angle 6 between the normal to the interface and the normal to the cell face (Fig. 8). For an interface aligned with the cell face (9 = 0), the bounded downwind scheme is used, while for an interface perpendicular to the cell face, the upwind scheme is used. For an interface with 6 between these two limits,/(0) is chosen to be V cos 6 and the blending formula is given by... 

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