Hi,

Why would someone use QUICK, TVD, ENO, etc for the linear advection equation. Why is ENO considered best, TVD next and finally QUICK? How is this know mathematically...or is it based on empirical ( results from simulation etc) data?

For sufficiently small values of "viscosity", the linear advection equation evidences a sharp gradient in its solution, the occurence often present in real-life problems. If you do not pack a large (very large) number of nodes inside of this high gradient area, and use a "non-optimized" fe, fd or fv discretization technique (QUICK for example) your solution will be polluted by dispersion errors (you'll see oscillations) signaling the inability of the mesh to resolve the sharp gradient (in many codes this is masked by adding artificial diffusion, which comes in all kinds of shapes and forms). Using a large number of nodes gets expensive in 3D and the location of the gradient region in not necessarily known in advance. TVD, ENO, WENO, compact difference fd, optimal h-p fe, tws fe just to name a few are designed in an attempt to eliminate dispersive errors in the numerical solution without introducing excess diffusion in the process.