Fundamentals of the Finite-volume Method

The grid node is surrounded by four neighboring nodes N, S, W and . 

A priori, neither the value of O nor the values of density and velocity are known at the faces of the control volume. They have to be determined via interpolation from their values at neighboring nodes. A simple approximation would be 

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. 

In order to increase the accuracy of the approximation to the convective term, not only the nearest-neighbor nodes, but also more distant nodes can be included in the sum appearing in Eq. (37). An example of such a higher order differencing scheme is the QUICK scheme, which was introduced by Leonard [82]. Within the QUICK scheme, an interpolation parabola is fitted through two downstream and one upstream nodes in order to determine O on the control volume face. The un- 

The QUICK scheme has a truncation error of order h. However, similarly as in the case of the central differencing scheme, at high flow velocities some of the coupling coefficients of Eq. (37) become negative. 

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