The numerical methods for the solution of the compressible Euler and Navier-Stokes equations can be classified by the mesh they use as structured grid methods, unstructured grid methods, Cartesian grid methods, and gridless methods. Each of these methods, advocated, promoted, developed, and used by their respective supporters, has its own advantages and disadvantages. The structured grid methods have a disadvantage in mesh generation for complex geometries. The main advantage of the unstructured grid methods is the ease of grid generation for complex configurations. However, the computational costs and memory requirements are generally higher than their structured grid counterparts. The advantages of the Cartesian grid methods include ease of grid generation, lower computational storage requirements, and significantly less operational count per cell.