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Next: 2.19 Shapiro Filter
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2.18 Comparison of advection schemes
Table 2.2:
Summary of the different advection schemes available in MITgcm.
``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
The code corresponds to the number used to select the corresponding
advection scheme in the parameter file (e.g., tempAdvScheme=3 in
file data selects the order upwind advection scheme
for temperature).
Advection Scheme |
code |
use |
use Multi- |
Stencil |
comments |
|
|
A.B. |
dimension |
(1 dim) |
|
centered order |
2 |
Yes |
No |
3 pts |
linear |
order upwind |
3 |
Yes |
No |
5 pts |
linear/ |
centered order |
4 |
Yes |
No |
5 pts |
linear |
order DST |
30 |
No |
Yes |
5 pts |
linear/, non-linear/v |
order Flux Limiters |
77 |
No |
Yes |
5 pts |
non-linear |
order DST Flux limiter |
33 |
No |
Yes |
5 pts |
non-linear |
|
Figs. 2.14, 2.15 and
2.16 show solutions to a simple diagonal
advection problem using a selection of schemes for low, moderate and
high Courant numbers, respectively. The top row shows the linear
schemes, integrated with the Adams-Bashforth method. Theses schemes
are clearly unstable for the high Courant number and weakly unstable
for the moderate Courant number. The presence of false extrema is very
apparent for all Courant numbers. The middle row shows solutions
obtained with the unlimited but multi-dimensional schemes. These
solutions also exhibit false extrema though the pattern now shows
symmetry due to the multi-dimensional scheme. Also, the schemes are
stable at high Courant number where the linear schemes weren't. The
bottom row (left and middle) shows the limited schemes and most
obvious is the absence of false extrema. The accuracy and stability of
the unlimited non-linear schemes is retained at high Courant number
but at low Courant number the tendency is to loose amplitude in sharp
peaks due to diffusion. The one dimensional tests shown in
Figs. 2.12 and 2.13 showed this
phenomenon.
Finally, the bottom left and right panels use the same advection
scheme but the right does not use the multi-dimensional method. At low
Courant number this appears to not matter but for moderate Courant
number severe distortion of the feature is apparent. Moreover, the
stability of the multi-dimensional scheme is determined by the maximum
Courant number applied of each dimension while the stability of the
method of lines is determined by the sum. Hence, in the high Courant
number plot, the scheme is unstable.
With many advection schemes implemented in the code two questions
arise: ``Which scheme is best?'' and ``Why don't you just offer the
best advection scheme?''. Unfortunately, no one advection scheme is
``the best'' for all particular applications and for new applications
it is often a matter of trial to determine which is most
suitable. Here are some guidelines but these are not the rule;
- If you have a coarsely resolved model, using a
positive or upwind biased scheme will introduce significant diffusion
to the solution and using a centered higher order scheme will
introduce more noise. In this case, simplest may be best.
- If you have a high resolution model, using a higher order
scheme will give a more accurate solution but scale-selective
diffusion might need to be employed. The flux limited methods
offer similar accuracy in this regime.
- If your solution has shocks or propagating fronts then a
flux limited scheme is almost essential.
- If your time-step is limited by advection, the multi-dimensional
non-linear schemes have the most stability (up to Courant number 1).
- If you need to know how much diffusion/dissipation has occurred you
will have a lot of trouble figuring it out with a non-linear method.
- The presence of false extrema is non-physical and this alone is the
strongest argument for using a positive scheme.
Next: 2.19 Shapiro Filter
Up: 2. Discretization and Algorithm
Previous: 2.17.4 Multi-dimensional advection
Contents
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