Role of coordinates in computational fluid dynamics

Computational fluid dynamics uses large scale numerical computation to solve problems of fluid flow. It turns out that the numerical solution for a given flow depends on the coordinates (grid) used to compute the flow. The commonly used Eulerian and Lagrangian coordinate systems both have advantages and drawbacks. In this paper, we first discuss the role of coordinates in computational fluid dynamics regarding the questions of: (a) conservation form partial differential equations; (b) numerical resolution of contact discontinuities; (c) grid generation; and (d) grid orthogonality. We then introduce a unified coordinate system which combines the advantages of both Eulerian and Lagrangian system and beyond, while avoiding their drawbacks. Examples include a transonic flow past an airfoil and a two-fluids flow with shocks.