This is a sort of negative-leaning answer to the question about existence of software for your purpose.    There is quite a bit of history to the problem in prime characteristic, going back to isolated examples found in the 1970s by Mumford and his 1975 Ph.D. student W.L. Griffith Jr. (*Cohomology of Flag Varieties in Characteristic p*) which showed that the classical ideas could break down.    The rank 2 example $G_2$ lends itself to picture drawing and has been looked at in considerable detail.   See the recent updated preprint by Andersen and Kaneda <a href="http://front.math.ucdavis.edu/1107.3055">here</a>.  

Andersen's clever sheaf cohomology techniques (exploiting the Frobenius map) combined with my more speculative predictions tend to imply that the results depend heavily on Kazhdan-Lusztig theory for the affine Weyl group (of Langlands dual type).   Moreover, the non-vanishing of cohomology seems to involve the actual module structure, so dimensions appear only as a byproduct of the study of generic module filtrations crossing Weyl chamber walls.   The algebraic group of type $G_2$ already indicates how systematic but complicated the results will be in general, so any computational approach must take this case into account.  (The results for $A_2$ and $B_2$ worked out by Andersen following his 1977 MIT thesis *On Schubert Varieties in G/B and Bott's Theorem* are also subtle, but con't compete with the complexity of $G_2$ whose alcove geometry is richer.) 

ADDED: The problem arose in the setting of algebraic geometry, as seen in the thesis work mentioned above.   Seshadri wrote up his own version of the $SL_3$ case treated by Larry Griffith, in a typescript *Cohomology of line bundles on $SL_3/B$* (Tata Institute, September 28, 1976).  I learned about the problem from him the following spring at IAS and formulated my own tentative interpretation in a conference paper that summer.  Andersen recovered Griffith's results in a general setting in his 1979 Inventiones paper <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002095068">here</a>.  In particular, an extra non-vanishing $H^1$ has a unique simple submodule of specified highest weight.   But pinning down the dimension or formal character of this module takes more work, done first by Jantzen (before Kazhdan-Lusztig theory).   There may be shortcuts in small cases, but a general algorithmic approach to the flag variety of $SL_3$ gets complicated.