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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!

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  • $\begingroup$ I got a comment elsewhere. It seems all the numerical fluxes are approximate. Then I wonder whether any finite volume method is approximate, as numerical flux in DGTD actually is from finite volume method. $\endgroup$
    – Pu Zhang
    Jan 11, 2023 at 15:43
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    $\begingroup$ Note that you may use a numeric flux function which is based on the exact solution of an initial value problem, called "exact Riemann solution". Moreover, the Riemann problem theory is not restricted to constant initial value problems for hyperbolic problems, only. The "generalized" Riemann problem theory may also be used for hyperbolic-parabolic problems. $\endgroup$
    – ConvexHull
    Jan 11, 2023 at 17:40
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    $\begingroup$ No scheme is "exact" in the sense that it gives you the exact solution on a mesh with finite mesh size. All of these schemes only give you approximations to the solution, and the question then is "does the approximate solution converge to the exact one", "how accurate is the approximate solution", and "by which convergence order does convergence happen". Different fluxes will have different answers to these questions. $\endgroup$ Jan 11, 2023 at 18:07
  • $\begingroup$ Thanks, @ConvexHull! So in an inverse sense a rigorous numeric flux can be found by solving generalized Riemann problem. But does this approach lead to a practical method? Are the centered and upwind fluxes derived from the theory of generalized Riemann problem? $\endgroup$
    – Pu Zhang
    Jan 12, 2023 at 0:18
  • $\begingroup$ @WolfgangBangerth, thank you for helping improving the formulation of my problem! That's what I meant. When using different fluxes, the solution would converge to different results (if the solution converges). Are there answers for the most used fluxes, like the centered and upwind? $\endgroup$
    – Pu Zhang
    Jan 12, 2023 at 0:24

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