Variable grid scheme for discontinuous grid spacing and derivatives

A Crank-Nicolson type finite-difference scheme is developed for solving boundary layer flows on arbitrary grids and with jumps in viscosity and density. The method is applied to the similar equations and two approaches are obtained depending upon the linearization of terms. One of these approaches can be developed from the box scheme formulation. In some cases, difference relations for derivatives are those obtained in the variable grid scheme developed previously. Numerical solution verify that the difference techniques have second-order behavior as the grid system is refined. A wall velocity gradient relation is determined which gives second-order accuracy for all grids considered.