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

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Old   April 20, 2012, 09:13
Question How L-norms are used to study stability of a numerical scheme?
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Ravindra Shende
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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.
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Old   April 20, 2012, 11:08
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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.
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Old   April 20, 2012, 14:35
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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
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Old   April 20, 2012, 16:48
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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.
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Old   April 21, 2012, 04:02
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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.
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Old   April 21, 2012, 07:43
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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.
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Old   April 21, 2012, 13:36
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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.
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Old   April 22, 2012, 17:15
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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.
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