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I'm part of a team trying to generalize a 1D Advection-Diffussion-Reaction code we inherited by extending it to 2D by using dimensional splitting, i.e. solving advection and diffusion for x and y separately with 1D schemes.

Diffusion was (relatively) painless but there are issues with advection.

Originally the code had a flux limited, third order upwind-biased scheme for 1D advection which worked like a charm in the 1D case. However, when the same scheme is extended to 2D, crazy instabilities develop. What supposed to be a radial advection-diffusion profile wrinkles in the transverse direction and shoots a "jet-like" instability:

[Instability[1]

However when I use a first order upwind scheme I do get a nice, symmetric radial profile. The problem with the first order upwind scheme solution is the excessive false diffusion which the higher order scheme does suppress. But the higher order scheme is even more problematic because of the aforementioned instability.

I was wondering if higher order schemes in general are a bad idea in a multidimensional case? Are multidimensional problems more succeptible to "oscillations" caused by high order schemes? What about flux limiters, should 1D flux limiters prevent this oscillations from happening (in my situation, it seems the flux limiter for the third order case doesn't help)? Should I stick to lower order schemes even if they bring false diffusion? This third order scheme I was using is also flux limited, but the flux limit doesn't stop the instabilities...

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    $\begingroup$ If you decrease the time step by, say, a factor of 10 or 100 do you see the same behavior? $\endgroup$ – Rhys Ulerich Aug 4 '15 at 11:25
  • $\begingroup$ I could try although it's already really small and would get much slower (and its already slow), but I don't think its that. I tried a 2nd order flux limited scale and its stable and there's no jet thing. I was wondering if I could use a FCT scheme to limit the 3rd order flux instead of a 1D TVD limiter and maybe it will "limit" correctly the transverse part. $\endgroup$ – mathdummy Aug 4 '15 at 19:49
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    $\begingroup$ Higher order schemes in multiple dimensions are certainly not always a bad idea. However, split schemes can develop instabilities even if the individual steps are all stable (see e.g. 4.3.3 in springer.com/us/book/9781441964113) -- may be this is what happens here? $\endgroup$ – Daniel Aug 4 '15 at 21:29
  • $\begingroup$ I think what happens is that the front lags in the transverse direction which "wrinkles" the interface and creates pressure gradients which makes the system blow up. Maybe using a multidimensional Flux Corrected Transport will guarantee monoticity so that it doesn't blow up? $\endgroup$ – mathdummy Aug 5 '15 at 22:29
  • $\begingroup$ This would suggest that the problem goes away if you set it up in a way that the front travels in a direction aligned with an axis? Is this possible? $\endgroup$ – Daniel Aug 6 '15 at 7:08

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