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May 12, 2001, 07:56 
Convergence Criterion

#1 
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For steady inviscid compressible calculations using Euler equations, what is the convergence criterion that is usually used ?


May 12, 2001, 12:26 
Re: Convergence Criterion

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(1). If the CFD solution took only microsecond to converge, then you would not be asking this kind of question. It is a good question. And It is a common question. It is mainly related to the slow convergence of CFD solutions, rather than the convergence criterion. (2). For algebraic equation, when the solution converges, the solution should remain the same. If it is still changing, then it is not converged solution. (3). If the equation is 1D, then you can wait for a while to see if the solution is still changing. For 2D, this is also possible, but in 3D, it becomes very difficult. (4). The main reason is: we don't know how to get the converged solution. In almost all cases, the code is not smart enough to zoom in the converged solution. In 3D, if you have 100,000 points or cells, and each cell contain basic unknows of u,v,w,p,rho,e, then you will have a total of 600,000 unknowns. In the iterative process, it represents 600,000 degree of freedom. So, it is not easy to get a solution which will stay the same. (5). Your question is really: if I don't know whether the solution is the converged solution, can I still use the solution. It is equivalent to: if I don't know whether the gun is accurate enough to kill my enemy, should I fire my gun at my enemy? (6). The answer to your question is: it depends on whether the solution is going to be useful to you. (that determines when to stop the iteration)


May 14, 2001, 00:18 
Re: Convergence Criterion

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Hi, for convergence of the nonlinear iterations (as opposed to grid and time step convergence of the numerical scheme) a fairly good rule of thmub is to reduce the residuals 4 orders of magnitude. Some code manuals will say 3 orders is good enough and for academic work people often reduce them down to 78 orders or to the roundoff of the machine in single precision. The reduction of the residual can be shown to be a good indicator of the level of error in the current iterate of the nonlinear iterations BUT the two are NOT the same. So 4 orders of reducion will USUALLY give solutions which have 34 significant fugures of accuracy.
A very good discussion of the is and some examples are given in Ferziger and Peric's text: Computational Methods for Fluid Dynamics. Regards, Duane 

May 14, 2001, 02:17 
Re: Convergence Criterion

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I will be pleased to know the availability of recent text books containing example codes for the finite element analysis of incompressible fluid flow in 'c'/'c++' language. Regards.


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