A unified coordinates approach to computational fluid dynamics

A unified coordinate system is introduced for computational fluid dynamics, in which the grid moves with velocity hq,q being fluid velocity. It includes the Eulerian coordinates as a special case when h=0 and the Lagrangian when h=1. By suitably choosing the free function h-h=1 for 1D flow, h chosen to preserve grid angles for 2D flow, and to preserve grid skewness for 3D flow—the unified coordinate system is shown, in a large number of examples involving the Euler equations, to be superior to both Eulerian and Lagrangian ones in resolving flow discontinuities: shocks and especially slip lines. This approach is also successfully extended to shallow water waves and viscous flow.