
[Sponsors] 
December 6, 2011, 10:47 
How to determine the order from numerical experiments

#1 
New Member
Join Date: Dec 2010
Posts: 14
Rep Power: 6 
Hi!
I'm a relative rookie in the field of CFD and I have a basic question: how do you determine the order of a method from experimental results. I've seen in several papers that it is determined by using a certain norm (1,2 or inf), the grid spacing is then usually halfed in sequential steps, but then based on the Lnorm, how do you determine the order? Here is an example for a 1D domain. How are the values in the last column obtained? dx L2error Order dx = 1/ 40 0.18*10−3 dx = 1/ 80 0.13*10−4 3.80 dx = 1/160 0.86*10−6 3.90 dx = 1/320 0.56*10−7 3.96 dx = 1/640 0.35*10−8 3.98 Thank you very much! 

December 6, 2011, 16:41 

#2 
New Member
Jeff
Join Date: Aug 2010
Posts: 11
Rep Power: 6 
Check out the ASME V&V20 standard for CFD validation or search for the various papers and books by Roache on V&V for a cookbook methodology.


December 6, 2011, 18:22 

#3 
Senior Member
cfdnewbie
Join Date: Mar 2010
Posts: 551
Rep Power: 11 
Assume that the error of your approximation has the following form:
E1=c1*h1**p, where c1 is a constant, h1 is your current grid spacing and p is the order of your approximation (for h>0). Now consider another grid spacing h2: E2=c1*h2**p, with p=p, assume that c1 is due to the formulation and a constant factor now: E1/E2 = h1**p / h2**p Take the Log to get log (e1/e2) = p * log ( h1/h2) solve for p! so try it out with your example, e.g. the first pair: dx = 1/ 40 0.18*10−3 dx = 1/ 80 0.13*10−4 3.80 just did the math, came up with p=3.795, just try it! basic idea behind it is: assume error has polynomial shape with constant p, pick two grid spacings and associated errors and fit a curve thru them. slope gives order.... 

December 7, 2011, 01:50 

#4 
New Member

The order basically tells you how fast the error decreases as you refine your mesh. Firstorder convergence means that if you go from cell spacing of let's say 2mm to 1mm, your error will also decrease by a factor of 2. Second order convergence would mean that the error went down by a factor of 4.
To determine the error, you do basically what you said. You come up with some method to calculate the error  this is the norm, the L2 norm is used quite a lot. Then calculate the norm for the baseline mesh. Then make the mesh finer by refining the mesh. Calculate the norm again. Repeat as much as you like. You will now have a set of "xy" points that you can graph and analyze to determine the convergence behavior. 

December 7, 2011, 03:57 

#5 
New Member
Join Date: Dec 2010
Posts: 14
Rep Power: 6 
Thank you all for your help! It's clear to me now.


Thread Tools  
Display Modes  


Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
How to compute UDS fourth order gradient  Emma66  FLUENT  1  April 27, 2010 05:44 
OpenFOAM  2nd order of discretization?  makaveli_lcf  OpenFOAM Running, Solving & CFD  0  November 12, 2009 08:15 
Higher order discretization on staggered grid  Chandra Shekhar  Main CFD Forum  9  January 27, 2005 17:31 
Standard for checking and testing numerical schemes?  X. Ye  Main CFD Forum  7  August 31, 1999 17:05 
New Books and Numerical Software  Eleuterio TORO  Main CFD Forum  0  December 18, 1998 13:41 