A scheme that uses an implicit system of finite-difference equations to obtain solutions of the equations governing the two-dimensional supersonic motion of an inviscid gas is described. The method relies on pseudo-viscosity in order to calculate shock waves. Compared with characteristics methods, pseudo-viscosity methods have certain advantages. For example, shock waves are calculated automatically without special procedures and pseudo-viscosity methods are easily generalised so that problems with three or more independent variables may be considered. Pseudo-viscosity methods have not been used extensively in the field of aerodynamics, partly because of the difficulty in obtaining sufficiently accurate solutions in the neighbourhood of a boundary. The main purpose of this paper is to show how this difficulty can be overcome. The problem of integrating the equations of motion when a boundary condition has to be satisfied on an arbitrary curve is considered. Streamlines are used as one of the independent variables so that the boundary curve is a coordinate curve, and the equations of motion are used in a form which leads to a simple procedure at the wall. For a given system of partial differential equations it is possible to introduce pseudo-viscous terms in many ways, not all of which are satisfactory. The results presented show that the method proposed in this paper is adequate. The calculated results are accurate and vary smoothly in the neighbourhood of the boundary.