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July 19, 2002, 19:18 
decrease in residual order and it's influence

#1 
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Hi, For the same test case (laminar flow over a flat plate) I'm using an explicit scheme and I'm facing an interesting event that I'm not able to understand. First as I set the CFL =0.5 at 3000 iterations I have 2 orders decrease in the residual. When I set the CFL= 0.3 I have 3 orders decrease in residual at 3000 iterations. However, the interesting point is that the solution seems much more converged at the higher CFL at 3000 iteration though the residual is 1 order higher that that of the CFL=0.3. What does the decrease in residual mean at this point??? I can't make sence of it?? If the decrease in residual measures the degree of convergence then theoretically the solution at the low CFL should look more converged.. Can anyone help? Thanks


July 22, 2002, 08:38 
Re: decrease in residual order and it's influence

#2 
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You really need to compare the absolute magnitude of the residual as well. If you're just looking at order of magnitude reduction from the initial point, then this is dependent on the magnitude of the initial residual value.
With a smaller CFL you will have a smaller residual and smaller changes in your solution. This may be what you're seeing. 

July 22, 2002, 18:16 
Re: decrease in residual order and it's influence

#3 
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Pete makes some important points. I repeat them:
(a) one must consider the absolute level of the residual norm, not the decrease from initial levels. (b) for an equation du/dt  r = 0, the discretized function value (du)_i at the spatial gridpoint "i" is dt times the residual function value at the same point. thus, using (du)_i as a measure of the residual of the steadystate equation scales the residual by a factor of the time step. the more appropriate measure is the residual itself: r_i = (du/dt)_i the relationship of the residual norm to the error gridfunction is a subtle one. i would add the following: (c) one must check the discretization error of the numerical scheme. some discretizations may involve error terms proportional to (dx/dt). the "steadystate" answer for such schemes depends on the courant number. (d) the two runs you made with differing courant numbers follow different relaxation paths to the steady state. the CN=0.3 case is like a more damped relaxation than the CN=0.5 case. after 3000 iterations, the CN=0.5 case has progressed more towards steady state almost everywhere due to different and greater evolution of transients. however, a small region, probably in the vicinity of the leading or trailing edge, may possess a large residual, dominating the residual norm. whereas in the CN=0.3 case, those flow conditions at the leading/trailing edge have not arisen, leading to lower residual norm, even though the errorfunction is of greater magnitude almost everywhere. keep in mind that the residual norm is a single number that summarizes the residual gridfunction. (e) for some problems (though this is unlikely in the case of the flat plate), there may be multiple "steadystate" solutions. which one you "converge" to may depend on the initial conditions and the CN and other parameters. presumably, this is not the case here. 

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