A Cartesian grid method for compressible flows to compute shock waves

The accuracy of the Cartesian grid method has been explored for the 2D compressible Euler equations. We prescribe wall boundary conditions at ghost points near embedded boundaries by using local symmetry conditions. We test two ghost point treatments for supersonic flow over a wedge. In the standard ghost point treatment, the numerical solution at the corresponding mirror points is interpolated either linearly or quadratically. The accuracy of our second order node-centered finite volume method is independent of a linear or quadratic interpolation. In a simplified ghost point treatment, we consider the closest grid point in y-direction as mirror points of the ghost points. The simplified ghost point treatment exhibits lower or comparable mass flow error than the standard ghost point treatment. Moreover, the Cartesian grid and the body-fitted grid methods are applied to supersonic flow over a circular arc airfoil. The comparison of these two methods depicts the requirement of a larger number of grid points for the Cartesian grid method than the body-fitted grid method.