# Points Per Wavelength

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 October 6, 2014, 11:48 Points Per Wavelength #1 New Member   Issac Mamed Join Date: Oct 2014 Posts: 1 Rep Power: 0 Hi, I am trying to find the grid points per wavelength PPW requirement to achieve a phase speed error less than 0.1 percent for sixth order non compact and compact centered approximations to a first derivative. I looked everywhere but cannot find the relationship between PPW and phase speed error and also how it relates to the Taylor Series expansions in the non compact and compact schemes. Please help me out. Thx Issac

October 6, 2014, 17:16
#2
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Filippo Maria Denaro
Join Date: Jul 2010
Posts: 3,640
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Quote:
 Originally Posted by issacmamed Hi, I am trying to find the grid points per wavelength PPW requirement to achieve a phase speed error less than 0.1 percent for sixth order non compact and compact centered approximations to a first derivative. I looked everywhere but cannot find the relationship between PPW and phase speed error and also how it relates to the Taylor Series expansions in the non compact and compact schemes. Please help me out. Thx Issac
I don't know if I got your question ... you want to get a formula to be compared to the exact first derivative of a function f(x) = f(k) exp(ikx).
Therefore you have the exact derivative

f'(x) = ik f(x)

and, on the other hand, a generic discrete operator Dn for the derivative will give you

Dn f(x) = ik_eff f(x)

being k_eff the resulting (can be also a complex number) modified wavenumber.
In conclusion, the local truncation error can be written as

L(x) = (d/dx - Dn) f(x) = i (k-k_eff) f(x)

and you want to minimize k-k_eff.

I suppose that one can start from the Nyquist theorem that says the minimum waveleght described on a grid needs at least of three grid points.
So you can start checking the error for several choices (that is for several k_eff)...

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