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Are there any good papers and or codes that couple discontinuous galerkin finite element solvers with Riemann solvers?

I need to explore coupling elliptic and hyperbolic problems but most splitting methods are ad hoc at best. Since I have a large amount of FEniCS code, I would like to just couple the Riemann solver with it. While a simple Roe solver would be a beginning, I'm looking for guidance on using more complicated methods.

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    $\begingroup$ All DG solvers for hyperbolic problems use Riemann solvers. Maybe you really want to ask about solving mixed hyperbolic-elliptic methods with DG methods? $\endgroup$ – David Ketcheson Nov 30 '11 at 17:47
  • $\begingroup$ @DavidKetcheson I see in your first comment to the question : >*All DG solvers for hyperbolic problems use Riemann solvers* I'm working on the code form Warburton for 1D euler. They do have slope limiters as is expected from most DG codes, but i am not sure of having seen a function that solves the discontinuous fluxes on the interfaces based on the flow direction. I am just a beginner in CFD, and have not come across a Riemann Solver code untill yet. I do have a code by Dr. Katate Masatsuka using Roe's approximate Riemann solver but is a FV code. I am not sure if there is a Riemann Solver imp $\endgroup$ – Suyash Sharma Jul 6 '16 at 2:14
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    $\begingroup$ If you have a new question, please ask it by clicking the Ask Question button. Include a link to this question if it helps provide context. - From Review $\endgroup$ – Christian Clason Jul 6 '16 at 6:19
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I suggest looking at the literature on DG methods for incompressible flow, which has the mixed hyperbolic-elliptic character you mention. There are a lot of approaches. This paper, for instance, even uses an exact Riemann solver. This one suggests using a discontinuous space for the hyperbolic part and a continuous one for the elliptic part.

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As with many high order methods, the accuracy of the scheme is often less sensitive to the Riemann solver. None of the DG papers for hyperbolic problems will actually be using averages, however. The most common choice is a Rusanov (aka. Local Lax-Friedrichs) flux, which is very simple if you have an upper bound for the fastest wave speed.

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    $\begingroup$ Good point. Complicated Riemann solvers are often overkill, especially if you have a high-order discretization. $\endgroup$ – David Ketcheson Dec 1 '11 at 19:58
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    $\begingroup$ @DavidKetcheson No, a good Riemann solver is not overkill, in particular those very complicated ones that are only a bit mor expensive than Lax-Friedrichs. High order of accuracy and solution error are not the same thing. Although they won't affect the order of accuracy, a good Riemann solver will significantly reduce your error, for a marginal increase in computational cost. $\endgroup$ – gnzlbg Aug 5 '12 at 13:49
  • $\begingroup$ @DavidKetcheson if by accuracy he means error yes it does. If he means order of accuracy then it does not. $\endgroup$ – gnzlbg Aug 5 '12 at 13:56
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    $\begingroup$ @gnzlbg In most cases, use of better Riemann solvers with high order methods is pretty much a wash. For example, this paper compares LxF to HLLC and finds that the latter has at best half the error on the same grid. Being a fifth order method, that is equivalent to refinement by 13%, which has similar incremental cost. Note also that the formally second order type A "WENO5" method is much more accurate than the second order TVD method. $\endgroup$ – Jed Brown Aug 5 '12 at 19:20
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    $\begingroup$ @JedBrown Indeed, I completely agree with you for HLL, HLLC, Roe... those are quite general fluxes, accurate, and also pretty heavy on computational cost. I meant, however, specialized fluxes like AUSM (Euler eqts. and NS for compressible flow), which are very cheap (almost same cost as LxF) and very accurate. Furthermore, one also has to considere how the time step scales with refinement ($\Delta t \approx O(h^2/p)$ I guess). Also, if you have discontinuities, h-refinement and low p won't cut it, you'll need a good flux. I can't however comment on ENO/WENO schemes, only DG. $\endgroup$ – gnzlbg Aug 9 '12 at 16:08

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