Transport: Non-diffusive, flux conservative initial value problems and ...

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concentration
concentration
3
2
1
0
3
2
1
0
60
75
t=0
90
15
105
30
120
45
0.09s staggered−leapfrog 1.32s mpdata (ncor=3, i3rd=1)
0
60 90 15 105 30 120 45
75
3
0.02s semi−lagrangian 15.85s pseudo−spectral
0
0 10 20 30
distance
3
2
1
2
1
t=0
0 10 20 30
distance
Figure 5.10: Comparison of solutions for advection of a tracer in a non-constant velocity
field all runs have 513 (or 512) grid points. Times in figures are for user time (cpu time) on
a SunUltra140e compiled with f77 -fast (a) Staggered-Leapfrog showing significant
numerical dispersion with courant number α = 0.5. 0.09 seconds. (b) 3rd-order 2 corrections
mpdata scheme. Lower dispersion but some diffusion (and 15 times slower) α = 0.5.
(c) Semi-Lagrangian scheme, 0.02 seconds α = 10. (d) iterative Pseudo-spectral scheme.
(512 points, α = .5) Excellent results but uber-expensive. Once the velocity is not constant
in 1-D, advection schemes become much more susceptible to numerical artifacts. Overall,
the semi-lagrangian scheme is far superior in speed and accuracy.
5.7 Pseudo-spectral schemes
Finally we will talk about Pseudo-spectral schemes which are more of related to
low-order finite difference techniques like staggered-leapfrog than to characteristics
approach of semi-lagrangian schemes. For some problems like seismic wave
propagation or turbulence the pseudo-spectral techniques can often be quite useful
but they are not computationally cheap.
The main feature of pseudo-spectral techniques is that they are spectrally accurate
in space. Unlike a second order scheme like centered space differencing which
uses only two-nearest neighgbors, the pseudo-spectral scheme is effectively infinite
order because it uses all the grid points to calculate the derivatives. Normally this