CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > General Forums > Main CFD Forum

How L-norms are used to study stability of a numerical scheme?

Register Blogs Members List Search Today's Posts Mark Forums Read

Like Tree2Likes
  • 1 Post By cfdnewbie
  • 1 Post By sbaffini

Reply
 
LinkBack Thread Tools Search this Thread Display Modes
Old   April 20, 2012, 09:13
Question How L-norms are used to study stability of a numerical scheme?
  #1
Member
 
Ravindra Shende
Join Date: Feb 2011
Location: Pune, India
Posts: 45
Rep Power: 15
Ravindra Shende is on a distinguished road
Greetings,

Can someone tell me in what way L1, L2 or infinity norm is used to study stability analysis of a numerical scheme?

I am trying to develop a code to solve compressible-Navier-Stokes equation using compact scheme for spatial discretization and runge-kutta scheme for temporal discretization. What other methods can be used for stability analysis of this scheme?

Thank you.
Ravindra Shende is offline   Reply With Quote

Old   April 20, 2012, 11:08
Default
  #2
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,839
Rep Power: 73
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
Quote:
Originally Posted by Ravindra Shende View Post
Greetings,

Can someone tell me in what way L1, L2 or infinity norm is used to study stability analysis of a numerical scheme?

I am trying to develop a code to solve compressible-Navier-Stokes equation using compact scheme for spatial discretization and runge-kutta scheme for temporal discretization. What other methods can be used for stability analysis of this scheme?

Thank you.
The general Holder norm is simply an application on the N-dimensional space that give you a "number". That means that you must first define an error (for example the exact minus the numerical solution) in vector N-dimensional space and apply a norm on it. The norm is studied in general for vanishing time and mesh steps.
FMDenaro is offline   Reply With Quote

Old   April 20, 2012, 14:35
Default
  #3
Member
 
Ravindra Shende
Join Date: Feb 2011
Location: Pune, India
Posts: 45
Rep Power: 15
Ravindra Shende is on a distinguished road
Quote:
Originally Posted by FMDenaro View Post
The general Holder norm is simply an application on the N-dimensional space that give you a "number". That means that you must first define an error (for example the exact minus the numerical solution) in vector N-dimensional space and apply a norm on it. The norm is studied in general for vanishing time and mesh steps.
Hello Filippo,

Thanks for your reply! Now, in addition to studying the norms for vanishing step sizes, I would like to know how can the norms be used in determining stability of a numerical scheme. My next question is about that.

I applied the scheme mentioned in my previous post to one dimensional linear wave equation (u_t + a * u_xx = 0) with Gaussian function as initial condition. I used cfl of 0.5.

I have attached a plot in this post which shows the L1, L2 and infinity norm of the error. The error is defined as the difference between exact solution and analytical solution.

The plots show that the L-norms are uniformly increasing with number of time iterations.

Now, from this plot, what can I say about the stability of this scheme?

This is the link to the plot.
http://dl.dropbox.com/u/56389861/norms%20of%20error.png

Last edited by Ravindra Shende; April 20, 2012 at 14:46. Reason: very large image size
Ravindra Shende is offline   Reply With Quote

Old   April 20, 2012, 16:48
Default
  #4
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,839
Rep Power: 73
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
a) The linear wave equation is u_t + a*u_x = 0, is a first order equation, not a second order.

b) the stability of a numerical scheme can be studied by the von Neumann analysis, the magnitude of the amplification factor must be less the unity.

c) The Lax theorem implies that a consistent and stable linear scheme must converge to the exact solution

Therefore, the stability in your case can be studied analitically, you have no reason to do a numerical case. You could use a norm on the error to verify the error slope and the accuracy order.
FMDenaro is offline   Reply With Quote

Old   April 21, 2012, 04:02
Default
  #5
Senior Member
 
cfdnewbie
Join Date: Mar 2010
Posts: 557
Rep Power: 20
cfdnewbie is on a distinguished road
Quote:
Originally Posted by Ravindra Shende View Post


Now, from this plot, what can I say about the stability of this scheme?

This is the link to the plot.
http://dl.dropbox.com/u/56389861/norms%20of%20error.png
you can assume that for the time given, the scheme is stable, since the error increases linearly with iteration number. That's what's to be expected for a stable scheme. If your scheme becomes unstable, expect a non-lineat (exponential) behaviour. However, you should do as the previous poster suggested: Do a von Neumann analysis, it will give you the CFL bounds for stability.
Ravindra Shende likes this.
cfdnewbie is offline   Reply With Quote

Old   April 21, 2012, 07:43
Default
  #6
Member
 
Ravindra Shende
Join Date: Feb 2011
Location: Pune, India
Posts: 45
Rep Power: 15
Ravindra Shende is on a distinguished road
Thank you for your replies!!!

Sorry for the error. I did mean the equation u_t + a*u_x = 0 and the plot is for this equation.

I used this equation just as a simple example to ask my question about the use of L-norms in determining the stability of a numerical scheme.

Now, von-Neumann stability analysis cannot be used for compressible-Navier-Stokes equations. So, can the concept of linear growth of L-norms of error be used as a stability criterion? What other methods are used to study the stability of a numerical scheme meant for compressible-Navier-Stokes equations?

Kind regards.
Ravindra Shende is offline   Reply With Quote

Old   April 21, 2012, 13:36
Default
  #7
Senior Member
 
sbaffini's Avatar
 
Paolo Lampitella
Join Date: Mar 2009
Location: Italy
Posts: 2,180
Blog Entries: 29
Rep Power: 39
sbaffini will become famous soon enoughsbaffini will become famous soon enough
Send a message via Skype™ to sbaffini
Dear Ravindra,

i suggest you to read:

LeVeque: Finite Volume Methods for Hyperbolic Problems - Chapter 8
Hirsch: Numerical Computation of Internal and External Flows - Chapter 7, 8, 9

where the concepts of norms, stability and convergence are fully clarified.

However, as you said, classical Von Neumann analysis (which, by the way, is stability in L2 norm) is not suitable for non linear problems. In that case (but this really is not my field) i think you have to move to more general concepts like Total Variation Bounding where, still, some specific norm is applied to some specific quantity. However, i don't think that there are general stability results concerning systems of non-linear equations like the compressible NSE.
sbaffini is offline   Reply With Quote

Old   April 22, 2012, 17:15
Default
  #8
Senior Member
 
sbaffini's Avatar
 
Paolo Lampitella
Join Date: Mar 2009
Location: Italy
Posts: 2,180
Blog Entries: 29
Rep Power: 39
sbaffini will become famous soon enoughsbaffini will become famous soon enough
Send a message via Skype™ to sbaffini
However, as said by cfdnewbie, as long as the error norms grow linearly (and not exponentially) you can conclude that there is no instability (it is just the buildup of error), no matter what system you are considering. The converse, of course, could not be true.
Ravindra Shende likes this.
sbaffini is offline   Reply With Quote

Reply

Tags
compact schemes, l-norms, stability analysis

Thread Tools Search this Thread
Search this Thread:

Advanced Search
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are Off
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
Numerical viscosity due to the MUSCL and HLL coulpled scheme sonsiest Main CFD Forum 0 May 23, 2011 15:37
problem about numerical scheme in LES. libin Main CFD Forum 4 July 1, 2004 04:32
the numerical scheme for LES. John S Main CFD Forum 2 March 14, 2004 07:52
Stability for Nonlinear Numerical Scheme Guo Main CFD Forum 3 February 12, 2001 12:21
numerical scheme ado Main CFD Forum 3 October 12, 2000 08:20


All times are GMT -4. The time now is 01:49.