Problems
235
Sx, Sketch an example for n = 2 and explain the significance of the change
in sign of the wavelength of the aliased wave.
2. Suppose that the spectral method is used to integrate a system including the
equation
8ifJ
a;+ " '+ifJx1/I=O,
and that the term ifJ (x, t) X (x, t) 1/1 (x, t) is to be evaluated using the transform technique . If K is the number of modes retained in the Fourier expansions for ifJ , X, and 1/1, derive an expression determining the minimum
number of grid points that must be present on the physical-space grid in
order to avoid aliasing error in the product ifJ X1/1 .
3. In all practical applications, finite Fourier transforms are computed using
the fast Fourier transform (FFf) algorithm (Cooley and Tukey, 1965). Suppose that the periodic spatial domain 0 x 2rr is discretized so that
2rr .
Xj = M]' where j = 1, . . . , M.
In order to be efficient, the FFf algorithm requires that M be the product of
small prime numbers . Maximum efficiency is obtained when M is apower
of 2. Thus , most FFf codes assume that M is an even number. When the
total number of grid points on the physical mesh is even, the finite Fourier
transform and inverse transform are given by the relations
and
ifJ(Xj,t)=
N
L ak(t)e
i kx j
k= -N+l
The 2N data points in physical space uniquely define 2N Fourier coefficients . However, in contrast to (4.14) , the wave number k = - N does not
appear in the expansion. Explain why the -N wave number is retained
in finite Fourier transforms when there is an odd number of points on the
physical mesh and dropped when the total number of points is even.
4. Solutions to the two-dimensional advection equation
81/1 + u 81/1 + v 81/1 = 0
8t
8x
8y
are sought in a domain that is periodic in both x and y . The velocity field is
nondivergent.
(a) Show that the domain integral of 1/13 is conserved by the exact solution
to the unapproximated goveming equations.
235
Sx, Sketch an example for n = 2 and explain the significance of the change
in sign of the wavelength of the aliased wave.
2. Suppose that the spectral method is used to integrate a system including the
equation
8ifJ
a;+ " '+ifJx1/I=O,
and that the term ifJ (x, t) X (x, t) 1/1 (x, t) is to be evaluated using the transform technique . If K is the number of modes retained in the Fourier expansions for ifJ , X, and 1/1, derive an expression determining the minimum
number of grid points that must be present on the physical-space grid in
order to avoid aliasing error in the product ifJ X1/1 .
3. In all practical applications, finite Fourier transforms are computed using
the fast Fourier transform (FFf) algorithm (Cooley and Tukey, 1965). Suppose that the periodic spatial domain 0 x 2rr is discretized so that
2rr .
Xj = M]' where j = 1, . . . , M.
In order to be efficient, the FFf algorithm requires that M be the product of
small prime numbers . Maximum efficiency is obtained when M is apower
of 2. Thus , most FFf codes assume that M is an even number. When the
total number of grid points on the physical mesh is even, the finite Fourier
transform and inverse transform are given by the relations
and
ifJ(Xj,t)=
N
L ak(t)e
i kx j
k= -N+l
The 2N data points in physical space uniquely define 2N Fourier coefficients . However, in contrast to (4.14) , the wave number k = - N does not
appear in the expansion. Explain why the -N wave number is retained
in finite Fourier transforms when there is an odd number of points on the
physical mesh and dropped when the total number of points is even.
4. Solutions to the two-dimensional advection equation
81/1 + u 81/1 + v 81/1 = 0
8t
8x
8y
are sought in a domain that is periodic in both x and y . The velocity field is
nondivergent.
(a) Show that the domain integral of 1/13 is conserved by the exact solution
to the unapproximated goveming equations.