Upwinding error

It can easily be shown that for the upwind scheme all coefficients a appearing in Eq. (37) are positive [81]. Thus, no unphysical oscillatory solutions are foimd and stability problems with iterative equation solvers are usually avoided. The disadvantage of the upwind scheme is its low approximation order. The convective fluxes at the cell faces are only approximated up to corrections of order h, which leaves room for large errors on course grids. 

The results of Equations 9"and 10 do not take account of any "background" SO2 and fine particle sulfur and selenium originating upwind of the Ft. Martin plant. This omission is not expected to be a major source of error, since any such corrections occur in both numerator and denominator of measurement ratios. In any... 

Several schemes and algorithms for solving the fluid dynamic part of the model have been published. This work has been concentrated on several items. Most important, one avoids using the very diffusive first order upwind schemes discretizing the convective terms in the multi-fluid transport equations. Instead higher order schemes that are more accurate have been implemented into the codes [62, 139, 140, 65, 105, 66[. The numerical truncation errors induced by the discretization scheme employed for the convective terms may severely alter the numerical solution and this can destroy the physics reflected... 

The flow is convection dominated and an essential feature of the method is to "upwind" the convected flux. It is assumed that the temperature of the flux crossing a boundary and "flowing" from node to node is defined by the temperature at the node which is upstream, and not by (say) an average value of the temperature at the two nodes. This gives excellent stability to the iterative solution of the equations for temperature, at some expense of increased truncation error. The method is robust and can accommodate reverse flow at any position in the film (which might occur from, say, recesses or hydrostatic supply ports). The domain can be set up and swept in the same way as an elliptic problem and, as observed, is strongly convergent. 

The finite volume method, which returns to the balance equation form of the equations, where one level of spatial derivatives are removed is the method of choice always for the pressure equation and nearly always for the saturation equation. Commercial reservoir simulators are, with the exception of streamline simulators, entirely based on the finite volume method. See [11] for some background on the finite volume method, and [26] for an introduction to the streamline method. The robustness of the finite volume method, as used in oil reservoir simulation, is partly due to the diffusive nature of the numerical error, known as numerical diffusion, that arises from upwind difference methods. An interesting research problem would be to analyse the essential role that numerical diffusion might play in the actual physical modelling process particularly in situations with unstable flow. In the natural formulation, where the character of the problem is not clear, and special methods applicable to hyperbolic, or near hyperbolic problems are not applicable, the finite volume method, in the opinion of the author, is the most trustworthy approach. 

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