The existence of modes of compressible fluid flow involving a separation of variables into a similarity solution in two dimensions and one-dimensional flow in the third is demonstrated. The numerical integration of such flows by a modified von Neumann–Richtmyer scheme is proposed, and the stability conditions investigated, showing that a generalized Courant–Friedrichs–Lewy condition is necessary. The inclusion of dissipation in the forms of artificial viscosity and thermal conduction into the model is discussed. The results of some test calculations are presented to demonstrate the behaviour of this model.