Second-Order Upwind

For a second-order upwind scheme, the convective flux is computed as:

(342)

where the face values and , are linearly interpolated from the cell values on either side of the face as follows:

(343)

(344)

where:

(345)

(346)

and and are the limited reconstruction gradients in cells 0 and 1, respectively.

The advantage of this scheme over the first-order upwind scheme is that it is nominally second-order accurate. However, the fact that the reconstruction gradients are limited helps to reduce local extrema and thus introduces more dissipation than a central differencing scheme. Clearly, the accuracy of this scheme will always be as good or better than the first-order upwind scheme. The downside is that, in some situations, the reduced numerical dissipation might result in poorer convergence properties than a first-order convection. Generally, this is an acceptable trade-off.