The comments made by Gartling are valid and indeed the reader will find of interest the improvement of results for a more advantageous boundary condition set. However the main point of the paper' is missed and needs re-emphasizing. This is that with a suitable upwinding parameter the accuracy of the approximation is improved irrespective of the boundary conditions. Indeed in the paper by Christie et a12 it is shown that for a one-dimensional problem the exact solution is available with such a parameter and one can hardly improve on this. In two-dimensional problems while the exact solution is not available an improvement of accuracy is very evident.
While the examples given by Gartling show that for gradient type downstream conditions adequate solutions are obtainable using standard (symmetric) Galerkin-Bubnov type weighting, this solution with unsymmetric, Galerkin-Petrov weighting using the optimal upwind parameter will be at least as good and very often better. As the convective term generates non-symmetric matrices in either case nothing is lost computationally and the possibility of improving accuracy is gained by using the procedure suggested.