The final result of this procedure is to give a set ofequations for the spectral
amplitudes ii(k). defined by (9), that are identical to (8) except for the replacement of the convolution sums by similarly truncated sums with
p, + y. = k, k 2 K or pa t y. = k, (Orszag, 1971a). The additional terms
entering the sums in ( 8 ) arc usually called "aliasing" terms. It has been
shown (Fox and Orszag, 1973) that the differences in the results obtained by
the present pseudospectral method and the spectral method are generally
negligible so long as either method gives an accurate solution of the equations of motion. Since the pseudospectral method may be implemented in
only nine Fourier transforms over (32)3 points per time step, it is roughly a
factor 2 more efficient than the spectral method and so has been used for
most of the simulations reported below.
Both the spectral and pseudospectral methods have been programmed for
a CDC-7600. The programs involve double buffering of data between small
core nicmory, large core memory, and disks. The spectral method requires
about 6 s per time step on the CDC-7600. while the pseudospectral code
requires only 3.2s per time step, both for the cutoffs K , = K 2 = 16.
K 3 = 32. Many of the critical internal loops of the program are written in
assembly language to avoid inefficiencies attributable to the Fortran
compiler, although further improvcrnents in the code should permit a
speedup of nearly 50 "/.
Resides the speed advantage of the pseudospectral method over the spectral method, the former has the great advantage that it applies with but the
most minor of modifications to problems involving more complicated physics, like chemical reactions or radiation. Since the expansions (7) are used
merely as an interpolatory tool in thc evaluation of derivatives, they may be
similarly used in the evaluation of these more complicated effects. Nevertheless, the pseudospectral method shares all the advantages of the spectral
method with regard to accuracy and, especially, efficiency improvement over
finitedifl'erence schemes (Orszag and Israeli, 1974). The expression of ( I ) in
rotation form is useful since it gives pointwise energy conservation.
3.2. N 0-S 1 ip Rout iiluries
satisfies
With no-slip boundary conditions applied at .Y, = 0. L , , the velocity
( I I )
v = 0
on .x, = 0. L , ,
v(s, + mL,, x2 + nL, , s3) = v(x)
inslead of (6). I t is no longer appropriate to use the Fourier expansions (7).
not just because 1 7 , = r2 = 0 at .Y, = 0. L , , but rather because imposition of
a Fourier series representation of the s3 dependence would induce Gibbs
phenomena at thc boundaries and result in slow convergence of the Fourier
series (Orszag. 1971a).
amplitudes ii(k). defined by (9), that are identical to (8) except for the replacement of the convolution sums by similarly truncated sums with
p, + y. = k, k 2 K or pa t y. = k, (Orszag, 1971a). The additional terms
entering the sums in ( 8 ) arc usually called "aliasing" terms. It has been
shown (Fox and Orszag, 1973) that the differences in the results obtained by
the present pseudospectral method and the spectral method are generally
negligible so long as either method gives an accurate solution of the equations of motion. Since the pseudospectral method may be implemented in
only nine Fourier transforms over (32)3 points per time step, it is roughly a
factor 2 more efficient than the spectral method and so has been used for
most of the simulations reported below.
Both the spectral and pseudospectral methods have been programmed for
a CDC-7600. The programs involve double buffering of data between small
core nicmory, large core memory, and disks. The spectral method requires
about 6 s per time step on the CDC-7600. while the pseudospectral code
requires only 3.2s per time step, both for the cutoffs K , = K 2 = 16.
K 3 = 32. Many of the critical internal loops of the program are written in
assembly language to avoid inefficiencies attributable to the Fortran
compiler, although further improvcrnents in the code should permit a
speedup of nearly 50 "/.
Resides the speed advantage of the pseudospectral method over the spectral method, the former has the great advantage that it applies with but the
most minor of modifications to problems involving more complicated physics, like chemical reactions or radiation. Since the expansions (7) are used
merely as an interpolatory tool in thc evaluation of derivatives, they may be
similarly used in the evaluation of these more complicated effects. Nevertheless, the pseudospectral method shares all the advantages of the spectral
method with regard to accuracy and, especially, efficiency improvement over
finitedifl'erence schemes (Orszag and Israeli, 1974). The expression of ( I ) in
rotation form is useful since it gives pointwise energy conservation.
3.2. N 0-S 1 ip Rout iiluries
satisfies
With no-slip boundary conditions applied at .Y, = 0. L , , the velocity
( I I )
v = 0
on .x, = 0. L , ,
v(s, + mL,, x2 + nL, , s3) = v(x)
inslead of (6). I t is no longer appropriate to use the Fourier expansions (7).
not just because 1 7 , = r2 = 0 at .Y, = 0. L , , but rather because imposition of
a Fourier series representation of the s3 dependence would induce Gibbs
phenomena at thc boundaries and result in slow convergence of the Fourier
series (Orszag. 1971a).