238
4. Series-Expansion Methods
is aliasing error affecting the stability and accuracy of the pseudospectral
solution?
(b) Repeat the preceding simulations using l:i.x = 1/128 and 128 Fourier
modes. Show the solutions at t = 0.40 in the subdomain 0 :s x :s !'
o :s 1/1 :s 1. How seriously is aliasing error degrading the stability and
accuracy of the pseudospectral solution?
(c) Why is there an improvement in the pseudospectral solution between
the simulations in (a) and (b)?
(d) If the spatial resolution is increased to 256 Fourier modes, both the
spectral and pseudospectral solutions become unstable. Why? Devise a way
around this instability and obtain an approximation to the solution at t =
0.40. Again plot this solution on the subdomain 0 :s x :s !' 0 :s 1/1 :s 1.
14. Solutions to the coupled advectionlchemical reaction equations
aljJ
aljJ
-
at
+c- =ljJ1/I,
ax
-
a1/l
+ c - = -ljJ1/I,
a1/l
at
ox
are to be obtained using the Galerkin finite-element approach. Assume that
the expansion functions are chapeau functions and that the approximate
expressions for 1/1 and ljJ are
Using (4.89) we know that the first equation will have the form
-
l:i.x (da
--+4-+-j +1
da]
da j _ \ ) +c
(ai+\ - aj _ \ ) =X ,
6
dt
dt
dt
2
where X represents the Galerkin approximation to ljJ1/I Evaluate X in terms
ofthe expansion coefficients ak and bi,
15. Determine the Galerkin finite-element approximation to 1/Ia1/llax using
chapeau expansion functions. Show that the result is identical to the conservative finite-difference operator appearing in (3.120).
16. *Compute solutions to Problem 13 of Chapter 3 using the spectral and pseudospectral methods. Use the same numerical parameters specified in that
problem except choose l:i.t so that the Courant number based on the maximum wind speed for the shortest wavelength retained in the spectral truncation is 0.3. Do not use any type of smoother. Use leapfrog time-differencing,
taking a single forward step to obtain the solution at the first time level.
(a) Obtain solutions using 64 Fourier modes to approximate 1/1 and c(x) .
Show yourresults at t = 1.5 and 3.0 as directed in Problem 13 of Chapter 3.
Also show the two solutions at some time when the pseudospectral method
is clearly showing some aliasing error. (Hint: this only happens for a limited
period of time during the integration.)
4. Series-Expansion Methods
is aliasing error affecting the stability and accuracy of the pseudospectral
solution?
(b) Repeat the preceding simulations using l:i.x = 1/128 and 128 Fourier
modes. Show the solutions at t = 0.40 in the subdomain 0 :s x :s !'
o :s 1/1 :s 1. How seriously is aliasing error degrading the stability and
accuracy of the pseudospectral solution?
(c) Why is there an improvement in the pseudospectral solution between
the simulations in (a) and (b)?
(d) If the spatial resolution is increased to 256 Fourier modes, both the
spectral and pseudospectral solutions become unstable. Why? Devise a way
around this instability and obtain an approximation to the solution at t =
0.40. Again plot this solution on the subdomain 0 :s x :s !' 0 :s 1/1 :s 1.
14. Solutions to the coupled advectionlchemical reaction equations
aljJ
aljJ
-
at
+c- =ljJ1/I,
ax
-
a1/l
+ c - = -ljJ1/I,
a1/l
at
ox
are to be obtained using the Galerkin finite-element approach. Assume that
the expansion functions are chapeau functions and that the approximate
expressions for 1/1 and ljJ are
Using (4.89) we know that the first equation will have the form
-
l:i.x (da
--+4-+-j +1
da]
da j _ \ ) +c
(ai+\ - aj _ \ ) =X ,
6
dt
dt
dt
2
where X represents the Galerkin approximation to ljJ1/I Evaluate X in terms
ofthe expansion coefficients ak and bi,
15. Determine the Galerkin finite-element approximation to 1/Ia1/llax using
chapeau expansion functions. Show that the result is identical to the conservative finite-difference operator appearing in (3.120).
16. *Compute solutions to Problem 13 of Chapter 3 using the spectral and pseudospectral methods. Use the same numerical parameters specified in that
problem except choose l:i.t so that the Courant number based on the maximum wind speed for the shortest wavelength retained in the spectral truncation is 0.3. Do not use any type of smoother. Use leapfrog time-differencing,
taking a single forward step to obtain the solution at the first time level.
(a) Obtain solutions using 64 Fourier modes to approximate 1/1 and c(x) .
Show yourresults at t = 1.5 and 3.0 as directed in Problem 13 of Chapter 3.
Also show the two solutions at some time when the pseudospectral method
is clearly showing some aliasing error. (Hint: this only happens for a limited
period of time during the integration.)