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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 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 $ 3^{rd}$ order upwind advection scheme for temperature).
Advection Scheme code use use Multi- Stencil comments
A.B. dimension (1 dim)
$ 1^{rst}$ order upwind 1 No Yes 3 pts linear/$ \tau$ , non-linear/v
centered $ 2^{nd}$ order 2 Yes No 3 pts linear
$ 3^{rd}$ order upwind 3 Yes No 5 pts linear/$ \tau$
centered $ 4^{th}$ order 4 Yes No 5 pts linear
$ 2^{nd}$ order DST (Lax-Wendroff) 20 No Yes 3 pts linear/$ \tau$ , non-linear/v
$ 3^{rd}$ order DST 30 No Yes 5 pts linear/$ \tau$ , non-linear/v
$ 2^{nd}$ order-moment Prather 80 No Yes    
$ 2^{nd}$ order Flux Limiters 77 No Yes 5 pts non-linear
$ 3^{nd}$ order DST Flux limiter 33 No Yes 5 pts non-linear
$ 2^{nd}$ order-moment 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    
$ 7^{nd}$ order one-step 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 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. 2.13 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 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 up previous contents
Next: 2.20 Shapiro Filter Up: 2. Discretization and Algorithm Previous: 2.18.4 Multi-dimensional advection   Contents
Copyright 2006 Massachusetts Institute of Technology Last update 2018-01-23