# Courant Number Violation for a Sink cell

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 April 8, 2023, 16:40 Courant Number Violation for a Sink cell #1 New Member   Dev K Join Date: Apr 2023 Location: USA Posts: 2 Rep Power: 0 I am writing some code to simulate overland waterflow. I am using an elevation grid obtained using a Digital Elevation Model. The direction of waterflow is decided by the max difference of the elevation between the a cell and its 8 neighbors. Water can flow out from a cell to only one of the 8 neighboring cells, while a single cell can receive water from multiple neighbors. In my grid, there are some 'sink' cells. These can be described as local minima, which have significantly lower elevation than its 8 neighbors. They can only receive the water, but cannot discharge water to any of its neighbors. Think of this cell as a lake. My model is an explicit model which requires Courant number to be less than 1 during each simulation step. During simulation, the CFL condition gets violated only at these 'sink' cells. I was wondering how critical is this violation, since this 'sink' cell can never transfer water to any of its neighbors? Even with a large water velocity it takes hours for this 'sink' to get filled. In such scenario, I achieve computational efficiency if I allow CFL condition to get violated for the sinks. Since the 'sink' cells can never discharge water to its neighbors, my understanding is that this violation may not be significant, and it won't affect overall model accuracy. Is my understanding correct? Appreciate any inputs you can provide. Thanks for your time. I am new to the field of CFD, hence please pardon if this is a very basic question.

 April 9, 2023, 06:17 #2 Senior Member     Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,169 Blog Entries: 29 Rep Power: 39 CFL is a dimensionless number whose significance is really dependent from the context. For example, in a fully implicit discretization of a linear convection based equation, one can show that it affects the accuracy but not the stability. So, we need to actually know the scheme and the equations you are using to really give any hint. Still, the mechanism you describe sounds non-linear to me. And, in addition, I expect such cells/lakes to eventually hold higher values with respect to their neighbors, if they keep receiving. Is this something that might then be affected by the high CFL? We really need more details here to help DevK likes this.

 April 9, 2023, 23:14 #3 Senior Member   Arjun Join Date: Mar 2009 Location: Nurenberg, Germany Posts: 1,281 Rep Power: 34 It may not cause problems for sink cells but its very likely cause problems for the cells connected to the sink cells if they are of similar sizes as sink cells. The reason is that courant criteria affects the way fluid transfers from one cell to other. So if it fails for sink then it fails for the neighbor too provided the cells of same sizes. DevK likes this.

 April 11, 2023, 09:32 #4 New Member   Dev K Join Date: Apr 2023 Location: USA Posts: 2 Rep Power: 0 Thank you Paolo and Arjun for your reply. I am using Sait-Venant's momentum equation to determine the velocity in a river channel. du/dt+u(du/dx)=-g(dh/dx)+gS-C(u**2)/h where h is water depth, u is velocity, S is slope, g is the gravity acceleration. C is sort of friction coefficient. The sink takes extremely large amount of time to fill in and start the outflow.

 Tags cfl, courant number, hydrology