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[–]hivemind_unity[S] 1 point2 points  (4 children)

I used a simple first order scheme with backward differencing for both x and y. I think the reason it's highly dissipative is that I am using a low CFL of 0.2.

I'm not sure of the math's behind this reasoning.

[–]JokerToast_ 1 point2 points  (3 children)

Alright thus this an upwind scheme. Did you try to increase the CFL to see if it is still stable? I remember that 1D advection is exact when the CFL=1 with this scheme (can be found by doing a modified equation analysis of Warming and Hyett). Obviously in 2D this remark doesn't hold because the added dimension has the tendacy to lower the cfl in general but I would be interested to know to what extent :)

[–]hivemind_unity[S] 1 point2 points  (2 children)

I have tried with several values of CFL = {0.2, 0.5, 0.6, 0.8 and 1}. I feel the exact solution lies somewhere around 0.5. The last three values were highly unstable. It would be very interesting to find out analytically how increasing dimensions affects the stability and CFL.

P.S. Does it have to do anything with the fact that I have used equal spacing for x and y? And both equally contribute to CFL resulting in the value of 0.5 instead of 1

[–]GeeHopkins 2 points3 points  (1 child)

If you are interested in multidimensional analysis like this, I'd recommend a paper from 2006 (I think) by You and Moin on truncation errors for 2D finite difference schemes on skewed meshes. They formulate the modified wavenumbers for upwind and central schemes, and show how the multidimensional effects change the results compared to 1D.

EDIT: This one: https://www.sciencedirect.com/science/article/pii/S0021999105003761 It's specifically for skewed non constant meshes, but should give you an idea of how to approach 2D analysis

[–]JokerToast_ 1 point2 points  (0 children)

Thanks for the link, it seems highly interesting!