In discontinuous Galerkin time domain (DGTD) method, a critical concept is the numerical flux that is used to link neighbouring elements. The numerical flux is however not unique. The popular choices include centered and upwind ones. I see in many articles and books describe their different numerical behaviours, but a problem has been hovering over my head for quite some time. Since distinct flux choices lead to DIFFERENT results, so rigorously speaking all these results CAN'T be correct. By "correct" I mean the solution faithfully satisfies the original partial differential equations. My question is: which flux choice is correct? (why do we need the wrong ones?) Or all the numerical fluxes are approximate? In the latter case, which is a better one?
Maybe my question is naive for people working with DGTD, the literature I read really did a good job to cause my confusion... Thanks for any comment!