194
4. Series-Expansion Methods
was selected such that
t1t
- max[o/(x , 0)] = 0.1 .
/)"x
x
The true solution to the inviscid problem gradually steepens around the point
x = ! and becomes discontinuous at t = (2rr)-1 (see Section 3.6.1), but the
viscous dissipation in (4.39) prevents the gradient at x = ! from collapsing to
a true discontinuity and gradually erodes the amplitude of the solution so that
o/(x, t)
0 as t
00. Fig. 4.3a shows the approximate solutions to (4.39)
obtained using a spectral truncation at wave number 64rr, which is equivalent to
a grid spacing of Sx = 1/64. Both the spectral and the pseudospectral solutions
have trouble resolving the steep gradient at x = ! and develop significant 2/)"x
noise . The amplitude of the 2/)"x ripples remains bounded in the spectral solution,
but aliasing error generates a rapidly growing instability in the 2/)"x component
of the pseudospectral solution.
Rather different results are, however, obtained if the same problem is repeated
with twice the spatia1 resolution. When the cutoff wave number is 128rr, the
pseudospectral method remains stable and actually generates a more accurate solution than that obtained with the spectra1 method. A elose-up comparison of the
two solutions over the subdomain 0 :s x :s ! appears in Fig. 4.3b. The 2/)"x
ripples in the spectral solution are of distinctly larger amplitude than those appearing in the pseudospectral solution. The pseudospectral solution remains stable because the rate at which viscous damping erodes a 2/)"x wave increases by
a factor of four as the spatial resolution is doubled from /)"x = 1/64 to 1/128,
and when Sx = 1/ 128, the rate of energy removal from the 2/)"x wave by viscous
damping exceeds the rate at which 2/)"x waves are amplified by aliasing error. The
superiority of the pseudospectral solution over the spectral solution in Fig. 4.3b
highlights the fact that although conservation of 114>112 implies stability, it does not
imply better accuracy.
The influence of aliasing error on accuracy is largely a matter of chance . Although it is certainly an error when the pseudospectral method misrepresents interactions between 2/)"x and 3/)"x waves as an aliased contribution to a 6/)"x wave,
it is also an error when the spectral method simply neglects the interactions between these same short waves, since the product of 2/)"x and 3/)"x disturbances
should properly appear in a 6/)"x/5 wave. In Burgers's equation, and in many
other fluid-flow problems, there is a cascade of energy to smaller scales. An accurate conservative scheme , such as the spectral method, replicates this down-scale
energy transfer except that the cascade is terminated at the shortest scales resolved
in the numerical simulation. In the absence of viscous dissipation, the spectral approximation to Burgers 's equation conserves energy, and the energy that cascades
down scale simply accumulates in the shortest resolvable modes. In order to simulate the continued cascade of energy into the unresolvable scales of motion, it
is necessary to remove energy from the shortest resolvable waves. The energy -
removal algorithm constitutes a parametrization of the influence of unresolved
short-wavelengths on the resolved modes and should be designed to represent the
true behavior of the physical system as closely as possible.
4. Series-Expansion Methods
was selected such that
t1t
- max[o/(x , 0)] = 0.1 .
/)"x
x
The true solution to the inviscid problem gradually steepens around the point
x = ! and becomes discontinuous at t = (2rr)-1 (see Section 3.6.1), but the
viscous dissipation in (4.39) prevents the gradient at x = ! from collapsing to
a true discontinuity and gradually erodes the amplitude of the solution so that
o/(x, t)
0 as t
00. Fig. 4.3a shows the approximate solutions to (4.39)
obtained using a spectral truncation at wave number 64rr, which is equivalent to
a grid spacing of Sx = 1/64. Both the spectral and the pseudospectral solutions
have trouble resolving the steep gradient at x = ! and develop significant 2/)"x
noise . The amplitude of the 2/)"x ripples remains bounded in the spectral solution,
but aliasing error generates a rapidly growing instability in the 2/)"x component
of the pseudospectral solution.
Rather different results are, however, obtained if the same problem is repeated
with twice the spatia1 resolution. When the cutoff wave number is 128rr, the
pseudospectral method remains stable and actually generates a more accurate solution than that obtained with the spectra1 method. A elose-up comparison of the
two solutions over the subdomain 0 :s x :s ! appears in Fig. 4.3b. The 2/)"x
ripples in the spectral solution are of distinctly larger amplitude than those appearing in the pseudospectral solution. The pseudospectral solution remains stable because the rate at which viscous damping erodes a 2/)"x wave increases by
a factor of four as the spatial resolution is doubled from /)"x = 1/64 to 1/128,
and when Sx = 1/ 128, the rate of energy removal from the 2/)"x wave by viscous
damping exceeds the rate at which 2/)"x waves are amplified by aliasing error. The
superiority of the pseudospectral solution over the spectral solution in Fig. 4.3b
highlights the fact that although conservation of 114>112 implies stability, it does not
imply better accuracy.
The influence of aliasing error on accuracy is largely a matter of chance . Although it is certainly an error when the pseudospectral method misrepresents interactions between 2/)"x and 3/)"x waves as an aliased contribution to a 6/)"x wave,
it is also an error when the spectral method simply neglects the interactions between these same short waves, since the product of 2/)"x and 3/)"x disturbances
should properly appear in a 6/)"x/5 wave. In Burgers's equation, and in many
other fluid-flow problems, there is a cascade of energy to smaller scales. An accurate conservative scheme , such as the spectral method, replicates this down-scale
energy transfer except that the cascade is terminated at the shortest scales resolved
in the numerical simulation. In the absence of viscous dissipation, the spectral approximation to Burgers 's equation conserves energy, and the energy that cascades
down scale simply accumulates in the shortest resolvable modes. In order to simulate the continued cascade of energy into the unresolvable scales of motion, it
is necessary to remove energy from the shortest resolvable waves. The energy -
removal algorithm constitutes a parametrization of the influence of unresolved
short-wavelengths on the resolved modes and should be designed to represent the
true behavior of the physical system as closely as possible.