Second and third order difference techniques for two-dimensional hyperbolic systems of conservation laws, with an application to transonic flow about an airfoil

The application of second and third order explicit difference techniques to hyperbolic systems of conservation laws is discussed. These difference methods are used to integrate the time-dependent differential equations describing inviscid fluid flow in two dimensions. Emphasis is placed on solving the problem on time-dependent transonic flow about a cusped airfoil. The interior of the unit disc is mapped conformally onto the exterior of the airfoil, and a unit circle is mapped homeomorphically onto the airfoil boundary. A polar grid is constructed in the unit disc and the fluid equations are solved on the grid. The method of characteristics is used to advance the solution at all boundaries of the domain. A cartesian grid in the physical plane is used to advance the fluid equations in a small neighborhood of the cusp. Steady state solutions accurate to second order are obtained as the asymptotic limit of the flow as it tends to infinity. Calculations containing shocks which have either a subsonic or supersonic free stream condition are presented.