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While looking into the difference between staggered and collocated grids, I came across an effect called $2\Delta x$-oscillations, which happen on non-staggered grids, but not on staggered grids. This phenomenon is explained in an article by Stelling (1984) on pages 101-103, but I don't exactly follow the reasoning. These spurious oscillations happen at water-height, are of order $\Delta x$, and seem to have something to do with boundary conditions. Could someone who knows about this topic clarify a little bit? For example, what are the "normal modes" they are looking for in this article?

EDIT: I'm particularly interested in knowing if these $2\Delta x$-oscillations are the wiggles behind a wave solution of the shallow water equations, because you can see that they're worse for the non-staggered grid. wiggles

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If you run the problem without accounting for the "spurious" modes, you will find that if the grid isn't fine enough somewhere to properly represent the solution. This can be due to BCs or other issues. Normal modes are the ones that don't jump "high/low/high" or vice-versa every neighboring node like, say Figures 2-18 through 2-21 in your link. Lots of things can cause this, but usually it's caused by not having enough mesh points in the problem area. These effects are often seen in boundary layers of both time-independent and time-dependent problems near the boundary in diffusion problems with central differencing, or, at least, that's the way I was taught. That they also come up due to time discretization method choice, was news to me.

There's usually a dimensionless condition that, when violated, leads to spurious oscillations. The so called "cell Peclet number" or "condition" is one that's common in the diffusion example with central differencing. The Peclet number is also used in thermo- and chemodynamics as a transport indicator, so you have to be a bit careful what you google for. Gresho and Lee have a classic paper on this in the context of the finite-element method: https://www.sciencedirect.com/science/article/abs/pii/0045793081900268.

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  • $\begingroup$ the thing I don't understand is how a grid that is not fine enough to represent the solution can exhibit MORE oscillation than the solution is supposed to have... Also, do you think these 2delta waves can explain the wiggles behind a wave solution or? $\endgroup$ Dec 14, 2021 at 11:56
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    $\begingroup$ @Bill Barth I think your explanation does not answer the question well. The topic rather concerns checkerboard oscillations which arises with non-consistent gradient approximations, e.g., also known from incompressible Navier-Stokes. See this video series to get a feeling where they come from: youtu.be/yqZ59Xn_aF8. Same, same but different. $\endgroup$
    – ConvexHull
    Dec 14, 2021 at 19:59
  • $\begingroup$ @ConvexHull Thank you for the link. Why would those oscillations only be visible after a big wave crest and why aren't they the same size everywhere? They seem to dampen out towards the "end" of the tail. $\endgroup$ Dec 15, 2021 at 9:52
  • $\begingroup$ The Peclet number is local and has the velocity and/or other local solution values in it. Conditions get worse as the solution and its derivative grow in absolute value. $\endgroup$
    – Bill Barth
    Dec 15, 2021 at 14:15

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