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Computational Fluid Dynamics
Lecture 9
Spectral Methods
The most accurate method for calculating spatial derivatives is to use Fast Fourier transforms.
Instead of being
() () ()
26
or
or
n
x
x
σ
∆∆∆
x
, the spectral, semispectral or Galerkin methods
converge to the exact solution faster than any algebraic value of
n
< ∞
.
As
x
∆
decreases, the rate of convergence is exponential. That does not mean however, that for
some finite value of
x
∆
that the spectral method will give a perfect solution.
It also fails for the
2
x
∆
wave, as did the compact schemes. i.e.
The disadvantages of the spectral method are:
1. Smaller time step required for stability.
fl
1
C
π
<
2. Slower computationally in the limit of large numbers of grid points.
Computational time, using FFT’s goes as
( )
log
NN
compared to finite difference schemes
that go as
.
()
N
In practice today, this is not a significant limitation.
3. Complex Geometries or Boundary Conditions are not as accessible as other methods.
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This note was uploaded on 06/08/2011 for the course EOC 6850 taught by Professor Slinn during the Spring '07 term at University of Florida.
 Spring '07
 Slinn

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