|October 6, 2014, 11:48||
Points Per Wavelength
Join Date: Oct 2014
Posts: 1Rep Power: 0
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.
|October 6, 2014, 17:16||
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 3,062Rep Power: 35
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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