During the development of computational methods that solve time dependent shock hydrodynamic problems, two underlying strategies have emerged that enable flow features to be resolved clearly. One, employ a numerical scheme of inherently high resolution, usually a second-order Godunov-type method. Two, locally refine the computational mesh in regions of interest. It has been demonstrated by Berger & Collela that a combination of both strategies is necessary if a solution of very high resolution is sought. The present study combines Roe's flux-difference splitting scheme with an adaptive mesh refinement algorithm developed from the ideas of Berger. The result being a general purpose scheme that can fully resolve complicated flows but which requires only modest computing power. The material in this thesis reflects three broad aims. First, to explain the methodology and intricacies of our scheme. Compared to non-adaptive methods our scheme is undeniably complicated, for it contains many elements which must be carefully co-ordinated. Second, to vindicate this complexity. To this end, computational results are presented which are comparable in resolution to Schlieren photographs, yet the calculations were performed on a small desktop workstation. Third, to give sufficient details of our implementation so as to allay the apprehensions of any person who might wish to code up the scheme.