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At the start, three and a half years ago, the finite fluid element method (Section 1) was the only one under consideration. Things went badly with that method, and progress with its implementation went far more slowly than we ever anticipated. As a result, by the end of the first year, a second method, which originally was developed for a check on results of the first, had become by far the more promising of the two. Most of this report (Sections 3-6) describes progress we have made with the application of biased differences (Section 2) to a variety of fairly difficult problems of numerical fluid mechanics. Finally, in the last six months of the period covered by this report, the major difficulties with finite fluid elements were overcome, so it became possible to begin a comparison of the two methods (Section 6). Preliminary indications are that both methods are reliable, and both are considerably more efficient than a third method with which they have been compared. Section 7 is a report of progress with a boundary integral method that is not closely related to the others except by being numerical, and Section 8 is a description of the kinds of graphical software we had to develop for interpretation of our numerical computations.