
[Sponsors] 
Is there such a thing as "method" independence? 

LinkBack  Thread Tools  Search this Thread  Display Modes 
February 14, 2020, 15:06 
Is there such a thing as "method" independence?

#1 
Member
Raphael
Join Date: Nov 2012
Posts: 35
Rep Power: 9 
I read somewhere that someone was claiming that you can have a mesh independent solution for some flow problem using the first order upwind scheme, but that it might still be giving a wrong result, because it was not "method" independent i.e. discretization scheme independent. Accordingly, this person claims that if you repeated a mesh independence using second order upwind scheme, you would get a different value at mesh convergence.
Is this true, and if so, how can first order upwind scheme provide mesh independent value that is still incorrect? 

February 14, 2020, 16:10 

#2 
Member
Join Date: Aug 2018
Posts: 57
Rep Power: 3 
If the numerical scheme is consistent, its solution converges to the solution of the PDE for h>0. A first order scheme will likely nerd a finer mesh than thesecond order one, but both will converge to thesame solution (for smooth problems!) or one has a bug.


February 14, 2020, 16:13 

#3 
Member
Raphael
Join Date: Nov 2012
Posts: 35
Rep Power: 9 
That is my understanding too....


February 14, 2020, 16:17 

#4  
Member
Raphael
Join Date: Nov 2012
Posts: 35
Rep Power: 9 
Quote:
Are you talking about discontinuities that would cause false diffusion to become significant, because mesh independence study should take care of that. If you are talking about discontinuities causing second order scheme to cause overshoots and undershoots and be worse than first order, then yes, that makes sense too. 

February 14, 2020, 16:39 

#5 
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 5,062
Rep Power: 54 
As addressed above, the Lax equivalence theorem ensures that a liner scheme that is consistent and stable converges towards the solution of the PDE, no matter about first, second or higher order of accuracy.
Maybe the issue is discussed in the framework of non linear problems? 

February 14, 2020, 16:41 

#6  
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 5,062
Rep Power: 54 
Quote:
"smooth problems" means that all the infinite terms (the derivatives) in the local truncation error are bounded and of unitary order of magnitude. 

February 14, 2020, 16:51 

#7  
Member
Raphael
Join Date: Nov 2012
Posts: 35
Rep Power: 9 
Quote:
Why would a nonlinear problem be different e.g. NS equations? 

February 14, 2020, 16:59 

#8  
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 5,062
Rep Power: 54 
Quote:
Linear scheme means that the process that produces the numerical solution is based on a linear transformation, that is f= A.x. For example the stationary time integration method, like FTUS, FTCS or other, can be written as f^n+1 = (A)^n+1 . f^0 You can make non linear a scheme also for a linear PDE. Of course, a non linear PDE generates a non linear algebric system 

Tags 
discretization scheme, first order upwind, second order upwind 
Thread Tools  Search this Thread 
Display Modes  


Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
[ICEM] Mesh Independence study  Carlos Modesto  ANSYS Meshing & Geometry  4  July 7, 2017 07:57 
Ahmed body 2D Questions about mesh independence, Y plus value and others  treviusss  FLUENT  0  July 6, 2017 08:52 
Grid Independence of Boundary Layer  Luigi_  STARCCM+  7  January 15, 2012 15:40 
Grid Independence of Boundary layer  Luigi_  Main CFD Forum  0  December 14, 2011 14:42 
Need for Mesh Independence Study  nickninevah  Main CFD Forum  6  October 15, 2010 18:25 