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Old   August 12, 2005, 13:58
Default convergence criteria
Andrew Hayes
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I am having trouble deciding on what kind of convergence criteria I should apply to my program. I have used MathCAD to develop a 2-D implicit finite difference program using the voticity/stream function method to get velocities. I am currently solving it for the flow in an enclosure with a moving top. In the program I can set determine how large my grid is, but I can't decide on how to set a convergence-criteria that will tell it when to stop iterating. Originally I was able to set the number of iterations. Now, I have it itereate until the absolute value of the center node's x-velocity, Ux, minus the previous iteration's Ux, is below a certain number, ie - .0001. I am not sure if that is a good method or not.

Any suggestions would be helpful.

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Old   August 12, 2005, 15:47
Default Re: convergence criteria
m. malik
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A criterion based on just the difference doesn't seem to be right. I use criterion of the follwing forms in variuos iterative problems: Sum(f_new - f_old)^2/Sum(f_new)^2 < a small value or Sum|f_new - f_old|/Sum|f_new| < a small value where Sum = sum on all the nodes

f = field variable new, old = current and previous iteration values
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Old   August 12, 2005, 19:24
Default Re: convergence criteria
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Whatever you do, don't just look at one point in the domain. If you're solving an equation f(x,y,...) = 0, determine the R.M.S. residual over the entire domain. Malik's equations provide an equivalent approach, but the exponent 2 should be inside the summation.

Apart from that: Your initial approach to fix a number of iterations was actually more robust. You should still impose an upper limit for the number of iterations, because you never know whether a set residual can even be reached. Always observe the residual, but don't make a fixed residual your only condition for termination.
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Old   August 13, 2005, 04:27
Default Re: convergence criteria
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check it out:
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