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 Annie February 7, 2002 22:44

Chebyshev grid

Dear all,

I am using chebyshev spectral method to sovle PDE. The computation domain is taken [a,b]. I use a uniform gird in physical domain. I map this mesh into [-1,1] with chebyshev grid when I calculate the derivatives. But it doesn't work. I have checked my program many times. Now I doubt whether this transformation can not be applied in the Chebyshev spectral method. But it is reasonable mathematically.

If anyone give some suggestions to me I will appreciate very much.

Thanks a lot!

Annie

 Patrick Godon February 8, 2002 12:05

Re: Chebyshev grid

Dear Annie,

While you map say [a,b] onto [-1,+1] you actually make a transformation of coordinates. So if y goes from a to b, x will go from -1 to +1. And you have now y(x). where

y(x)=a*(1-x)/2 + b*(1+x)/2

such that

y(x) = x*(b-a)/2 +a/2 +b/2

then when you derivate y, you must use the chain derivative rule.

df/dx = df/dy*dy/dx

or

df/dy= (df/dx)/(dy/dx)

where df/dx is the chebyshev derivative of the function f expanded as a series of chebyshev polynomials. in this case

dy/dx= (b-a)/2

Another concern might be the boundary conditions, since the Spectral Methods are extremely sensitive to (wrong) boundary conditions. So if you did use the derivative chaine rule and it does not work, you might be doing something wrong with the boundary conditions. The boundary conditions have to be imposed on the characteristic variables of the flow, and do not have to be superimposed (i.e. for a first derivative in space, use one BC a one boundary only; for a second derivative equation use one BC at each boundary, etc..).

I hope this help.

Do no hesitate to post more if you experience any trouble.

Cheers, Patrick

 Patrick Hanley, Ph.D. February 8, 2002 12:23

Re: Chebyshev grid

Annie,

Are you using Gauss-Lobatto points or a similar distribution? This would produce the desired accuracy.

Regards, Patrick Hanley, Ph.D. Aerodynamics Software http://www.hanleyinnovations.com

 Annie February 8, 2002 22:13

Re: Chebyshev grid

Dear Godon and Hanley,

Thank you for your kind help!

I do use Gauss-Lobatto points.

I need a uniform distribution of nodes in x in physical domain which is very important for me to go on my processure further.

The mapping used by me is y=(-2/pi*acos(x)+1)*(b-a)/2+(a+b)/2 if x belongs to [-1,1].

Unfortunately, dy/dx at the boundary is singular. It exhibits the well-known Runge phenomenon. In fact, I don't know why this phenomenon exists. I also don't know whether there is a good thansformation formulae which can avoid this phenomenon while help me obtain a uniform grid in physical domain.

Thank you again.

Annie

 Raj Bissessur February 11, 2002 08:00

Re: Chebyshev grid

If you check the transformation, make sure you have the Jacobian and the metrics right. If not, you are certain to obtain wrong answers. If you want I have the MIT subroutines for calculating the Gauss-Lobatto-Legendre quadrature points and weights. They are easy to use and quite frankly will save you a lot of time and trouble. They are written in Fortran, so let me know if you wish to have them. I used them for a simple PDE and then in a 2D NS calculation. But reading from what the other guys have replied to you, their suggestion seem plausible and the right way to go.

All the best,.

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