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2.18.4 Multi-dimensional advection

Figure 2.15: Comparison of advection schemes in two dimensions; diagonal advection of a resolved Gaussian feature. Courant number is 0.01 with 30$ \times $ 30 points and solutions are shown for T=1/2. White lines indicate zero crossing (ie. the presence of false minima). The left column shows the second order schemes; top) centered second order with Adams-Bashforth, middle) Lax-Wendroff and bottom) Superbee flux limited. The middle column shows the third order schemes; top) upwind biased third order with Adams-Bashforth, middle) third order direct space-time method and bottom) the same with flux limiting. The top right panel shows the centered fourth order scheme with Adams-Bashforth and right middle panel shows a fourth order variant on the DST method. Bottom right panel shows the Superbee flux limiter (second order) applied independently in each direction (method of lines).
\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-lo-diag.eps}}

Figure 2.16: Comparison of advection schemes in two dimensions; diagonal advection of a resolved Gaussian feature. Courant number is 0.27 with 30$ \times $ 30 points and solutions are shown for T=1/2. White lines indicate zero crossing (ie. the presence of false minima). The left column shows the second order schemes; top) centered second order with Adams-Bashforth, middle) Lax-Wendroff and bottom) Superbee flux limited. The middle column shows the third order schemes; top) upwind biased third order with Adams-Bashforth, middle) third order direct space-time method and bottom) the same with flux limiting. The top right panel shows the centered fourth order scheme with Adams-Bashforth and right middle panel shows a fourth order variant on the DST method. Bottom right panel shows the Superbee flux limiter (second order) applied independently in each direction (method of lines).
\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-mid-diag.eps}}

Figure 2.17: Comparison of advection schemes in two dimensions; diagonal advection of a resolved Gaussian feature. Courant number is 0.47 with 30$ \times $ 30 points and solutions are shown for T=1/2. White lines indicate zero crossings and initial maximum values (ie. the presence of false extrema). The left column shows the second order schemes; top) centered second order with Adams-Bashforth, middle) Lax-Wendroff and bottom) Superbee flux limited. The middle column shows the third order schemes; top) upwind biased third order with Adams-Bashforth, middle) third order direct space-time method and bottom) the same with flux limiting. The top right panel shows the centered fourth order scheme with Adams-Bashforth and right middle panel shows a fourth order variant on the DST method. Bottom right panel shows the Superbee flux limiter (second order) applied independently in each direction (method of lines).
\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-hi-diag.eps}}

In many of the aforementioned advection schemes the behavior in multiple dimensions is not necessarily as good as the one dimensional behavior. For instance, a shape preserving monotonic scheme in one dimension can have severe shape distortion in two dimensions if the two components of horizontal fluxes are treated independently. There is a large body of literature on the subject dealing with this problem and among the fixes are operator and flux splitting methods, corner flux methods and more. We have adopted a variant on the standard splitting methods that allows the flux calculations to be implemented as if in one dimension:

$\displaystyle \tau^{n+1/3}$ $\displaystyle =$ $\displaystyle \tau^{n} - \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n}) + \tau^{n} \frac{1}{\Delta x} \delta_i u \right)$ (2.201)
$\displaystyle \tau^{n+2/3}$ $\displaystyle =$ $\displaystyle \tau^{n+1/3} - \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3}) + \tau^{n} \frac{1}{\Delta y} \delta_i v \right)$ (2.202)
$\displaystyle \tau^{n+3/3}$ $\displaystyle =$ $\displaystyle \tau^{n+2/3} - \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3}) + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)$ (2.203)

In order to incorporate this method into the general model algorithm, we compute the effective tendency rather than update the tracer so that other terms such as diffusion are using the $ n$ time-level and not the updated $ n+3/3$ quantities:

$\displaystyle G^{n+1/2}_{adv} = \frac{1}{\Delta t} ( \tau^{n+3/3} - \tau^{n} )$ (2.204)

So that the over all time-stepping looks likes:

$\displaystyle \tau^{n+1} = \tau^{n} + \Delta t \left( G^{n+1/2}_{adv} + G_{diff}(\tau^{n}) + G^{n}_{forcing} \right)$ (2.205)

\fbox{ \begin{minipage}{4.75in} {\em S/R GAD\_ADVECTION} ({\em gad\_advection.F}... ...r $V$: {\bf vTrans} (local) \par $W$: {\bf rTrans} (local) \par \end{minipage} }

Figure 2.18: Muti-dimensional advection time-stepping with Cubed-Sphere topology
\resizebox{3.5in}{!}{\includegraphics{s_algorithm/figs/multiDim_CS.eps}}

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Next: 2.19 Comparison of advection Up: 2.18 Non-linear advection schemes Previous: 2.18.3 Third order direct   Contents
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