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I have a cell centered cartesian grid and am trying to implement the flux inside the divergence term using numerical flux with a flux limiter. I found different formulas for MUSCL flux limiter, where one source seem to say one needs to use a different stencil depending on which way the quantity propagates at the face, and the other source just looks like a bit fancy finite difference stencil to me, with no "max" or if-statements needed. Is the propagation direction needed to use numerical flux? Can I just use velocity average of center and neighboring cell or do I need to differentiate flux with respect to the quantity flowing to get the propagation direction if so?

Further reading made me realize there are slope/gradient limiters and there are flux limiters, and there are Riemann solvers. I haven't found good (and free) information on this so do not know the details (yet). It does look like that most Riemann Solvers need the partials of the flux with respect to the quantity, so maybe the propagation direction dependence is actually a Riemann solver in guise.

So perhaps I should reconstruct two sided limits of the quantities at the interfaces using slope limiters, and then I should plug those values into the flux function, (and then add a half-time-step correction term $\frac{\Delta t}{2}\partial_t U$, for example with MUSCL-Hancock...? This does not seem to be done in PVM solvers though...?), and then I should plug those fluxes into a Riemann Solver, to get the one true flux at the interface. And then I should plug those solved fluxes into a time stepping scheme that uses Operator Splitting, to be able to incorporate source terms.

(Found a source using flux limiter. It used first ordered center (FORCE) scheme for the low order and Richtmyer scheme for high order. If I understood it that was a solution to the Riemann problem at the interface. It mentioned "upwind flux limiters" and "centered flux limiters" so I guess there might be some dependence on propagation direction in the flux limiter depending on scheme.)

(Found a source where it seems estimated convection speed (Roe averaged speed) plus minus estimated sound speed was used as input to HLL Riemann solver for Euler equations. Maybe something similar can be used in other fluid somulations.)

Emil
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  • Typically slope limiters are preferred according to my knowledge. Simple to implement, and understand the argument behind them and can be visualized nicely as well. – EMP Oct 27 '20 at 13:02

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Upwinding is not necessary. The best known and most-used limited central scheme is Nessyahu-Tadmor

Nessyahu, H. and Tadmor, E., 1990. Non-oscillatory central differencing for hyperbolic conservation laws. Journal of computational physics, 87(2), pp.408-463.

However a price is paid in that the contact discontinuity is badly smeared. Google Scholar will turn up the references

Philip Roe
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  • It seems to use staggered grid, how about upwind MUSCL scheme for cell centered collocated grids, is it possible to ignore propagation directions until solving the riemann problem, or do we need to take care to enter U_L and U_R with respect to propagation (i.e. does L stand for upstream and R for downstream or L for left and R for right?). This was my major painpoint, also in conjunction did not find anywhere if the limiters' $u_{1+1}$ could be taken to be the right neighbor, or had to be taken to be the downstream neighbor (in upwind MUSCL scheme for example). – Emil Oct 30 '20 at 06:35
  • I am sorry to give you these beginner's questions but guidance would be much appreciated! – Emil Oct 30 '20 at 06:43
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    @Emil The claim made is that N-T does not need to use Riemann solvers because of the staggered grid. A more accurate judgement might be that N-T is unable to use Riemann Solvers because of the staggered grid.

    The point of upwinding is to ensure the correct direction of information flow. Conventionally one uses a symmetric limiter F(x,y)=F(y,x) where x,y are gradients in the cells either side of the interface in question. It then does not matter which way the information is going. It does matter if the limiter is not symmetrical.

     – Philip Roe Oct 30 '20 at 21:53