A finite difference method of investigating the shock layer in the vicinity of the stagnation point in a two-dimensional supersonic flow past a circular cylinder is analyzed. This method differs from existing methods which have been used in the past for solving this type of problem in that it uses the properties along the stagnation streamline as initial conditions. However, the method can still be classified as one of the direct type. A potential flow solution along the stagnation streamline is used to approximate the initial conditions. The results are compared with existing solutions and available experimental data.