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2.19 Comparison of advection schemes
Table 2.2:
Summary of the different advection schemes available in MITgcm.
``A.B.'' stands for AdamsBashforth 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) 

order upwind 
1 
No 
Yes 
3 pts 
linear/
, nonlinear/v 
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 (LaxWendroff) 
20 
No 
Yes 
3 pts 
linear/
, nonlinear/v 
order DST 
30 
No 
Yes 
5 pts 
linear/
, nonlinear/v 
ordermoment Prather 
80 
No 
Yes 


order Flux Limiters 
77 
No 
Yes 
5 pts 
nonlinear 
order DST Flux limiter 
33 
No 
Yes 
5 pts 
nonlinear 
ordermoment Prather w. limiter 
81 
No 
Yes 


piecewise parabolic w. ``null'' limiter 
40 
No 
Yes 


piecewise parabolic w. ``mono'' limiter 
41 
No 
Yes 


piecewise quartic w. ``null'' limiter 
50 
No 
Yes 


piecewise quartic w. ``mono'' limiter 
51 
No 
Yes 


piecewise quartic w. ``weno'' limiter 
52 
No 
Yes 


order onestep method 
7 
No 
Yes 


with Monotonicity Preserving Limiter 












Figs. 2.15, 2.16 and
2.17 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 AdamsBashforth 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 multidimensional schemes. These
solutions also exhibit false extrema though the pattern now shows
symmetry due to the multidimensional 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 nonlinear 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. 2.13 and 2.14 showed this
phenomenon.
Finally, the bottom left and right panels use the same advection
scheme but the right does not use the multidimensional 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 multidimensional 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 scaleselective
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 timestep is limited by advection, the multidimensional
nonlinear 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 nonlinear method.
 The presence of false extrema is nonphysical and this alone is the
strongest argument for using a positive scheme.
Next: 2.20 Shapiro Filter
Up: 2. Discretization and Algorithm
Previous: 2.18.4 Multidimensional advection
Contents
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Massachusetts Institute of Technology 
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