4.2 The Spectral Method
185
such circumstances, the Galerkin requirement (4.9) becomes
- da; = - - L.... na
i
n
jlf
. k
c(x, t)e,(n- )x dx.
(4.24)
dt
21f n=-N
- l f
(4.25)
c(x, t) = L cm(t)e imx .
m=-N
Substitution of this series expansion into (4.24) gives
dak
dt
i
N
N
jlf .
e,(n+m-k)x dx ,
- l f
= - -
2
L....
'"" L....
'"" nc a
m n
1f n=-N m=-N
-
Although it may be possible to evaluate the integrals in (4.24) exactly for certain special flows, in most instances the computation must be done by numerical
quadrature. In a typical practical application, c(x, t) would be available at the
same spatial resolution as 'l/r(x, t); indeed, many models might include equations
that simultaneously predict c and 'l/r. Suppose, therefore, that c is given by the
Fourier series
N
(4.26)
which reduces , by the orthogonality of the Fourier modes, to
= - L incman.
m+n=1c
The notation below the summation indicates that the sum should be performed for
all indices n and msuch that Inl SN, Iml SN, and n +m= k.
Although the expression (4.26) is relatively simple, it is not suitable for implementation in large, high-resolution numerical models. The number of arithmetic
operations required to evaluate the time derivative of the kth Fourier coefficient
via (4.26) is proportional to the total number of Fourier coefficients, M == 2N +1.
The total number of operations required to advance the solution one time step
is therefore O(M
2 ) . On the other hand, the number of calculations required to
evaluate a finite-difference formula at an individual grid point is independent
of the total number of grid points. Thus, assuming that there are M points on
the numerical grid, a finite-difference solution may be advanced one time step
with just 0 (M) arithmetic operations. Spectral models are therefore less efficient
than finite-difference models when the approximate solution is represented by a
large number of grid points-or, equivalently, a large number of Fourier modes.
Moreover, the relative difference in computational effort increases rapidly with
increases in M. As a consequence, spectral models were limited to just a few
Fourier modes until the development of the transform method by Orszag (1970)
and Eliasen et al. (1970).
The key to the transform method is the efficiency with which fast Fourier transforms can be used to swap the solution between wave-number space and physical
space. Only O(M log M) operations are needed to convert a set of M Fourier coefficients, representing the Fourier transform of cP (x) , into the M grid-point values
185
such circumstances, the Galerkin requirement (4.9) becomes
- da; = - - L.... na
i
n
jlf
. k
c(x, t)e,(n- )x dx.
(4.24)
dt
21f n=-N
- l f
(4.25)
c(x, t) = L cm(t)e imx .
m=-N
Substitution of this series expansion into (4.24) gives
dak
dt
i
N
N
jlf .
e,(n+m-k)x dx ,
- l f
= - -
2
L....
'"" L....
'"" nc a
m n
1f n=-N m=-N
-
Although it may be possible to evaluate the integrals in (4.24) exactly for certain special flows, in most instances the computation must be done by numerical
quadrature. In a typical practical application, c(x, t) would be available at the
same spatial resolution as 'l/r(x, t); indeed, many models might include equations
that simultaneously predict c and 'l/r. Suppose, therefore, that c is given by the
Fourier series
N
(4.26)
which reduces , by the orthogonality of the Fourier modes, to
= - L incman.
m+n=1c
The notation below the summation indicates that the sum should be performed for
all indices n and msuch that Inl SN, Iml SN, and n +m= k.
Although the expression (4.26) is relatively simple, it is not suitable for implementation in large, high-resolution numerical models. The number of arithmetic
operations required to evaluate the time derivative of the kth Fourier coefficient
via (4.26) is proportional to the total number of Fourier coefficients, M == 2N +1.
The total number of operations required to advance the solution one time step
is therefore O(M
2 ) . On the other hand, the number of calculations required to
evaluate a finite-difference formula at an individual grid point is independent
of the total number of grid points. Thus, assuming that there are M points on
the numerical grid, a finite-difference solution may be advanced one time step
with just 0 (M) arithmetic operations. Spectral models are therefore less efficient
than finite-difference models when the approximate solution is represented by a
large number of grid points-or, equivalently, a large number of Fourier modes.
Moreover, the relative difference in computational effort increases rapidly with
increases in M. As a consequence, spectral models were limited to just a few
Fourier modes until the development of the transform method by Orszag (1970)
and Eliasen et al. (1970).
The key to the transform method is the efficiency with which fast Fourier transforms can be used to swap the solution between wave-number space and physical
space. Only O(M log M) operations are needed to convert a set of M Fourier coefficients, representing the Fourier transform of cP (x) , into the M grid-point values