This revision of [1], solicited by the Editors of Mathematical and Computer Modelling, gives an updated account of the Goldberg-Tadmor efforts to interpret the celebrated Gustafsson-Kreiss-Sundstrom stability theory in order to obtain simple stability criteria for finite difference approximations associated with the hyperbolic systems u(t) = Au-x + Bu + f in the quarter plane x greater than or equal to 0, t greater than or equal to 0.