Solution of governing equations for one-dimensional compressible unsteady flow has been performed traditionally using a homogenously distributed spatial mesh. In the resulting node structure, the internal nodes are solved by applying a shock capturing finite difference numerical method whereas the solution of the end nodes, which define the boundary conditions of the pipe, is undertaken by means of the Method of Characteristics. Besides the independent solution of every method, the coupling between the information obtained by the method of characteristics and the finite difference method is key in order to reach a good accuracy in gas dynamics modeling. The classical spatial mesh could provide numerical problems leading the boundary to generate lack of mass, momentum and energy conservation because of the interpolation methodology usually applied to draw the characteristics and path lines from its departure point at calculation time to the end of the pipe during the next time-step. To deal with this undesirable behavior, in this work a modification of the traditional grid including an extra node close to the boundary is proposed in order to explore its ability to provide numerical results with higher conservation fulfillment.