| 359 |
|
|
| 360 |
|
|
| 361 |
\section{Non-linear advection schemes} |
\section{Non-linear advection schemes} |
| 362 |
|
\label{sect:non-linear_advection_schemes} |
| 363 |
\begin{rawhtml} |
\begin{rawhtml} |
| 364 |
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| 365 |
\end{rawhtml} |
\end{rawhtml} |
| 628 |
\tau^{n+1/3} & = & \tau^{n} |
\tau^{n+1/3} & = & \tau^{n} |
| 629 |
- \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n}) |
- \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n}) |
| 630 |
+ \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\ |
+ \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\ |
| 631 |
\tau^{n+2/3} & = & \tau^{n} |
\tau^{n+2/3} & = & \tau^{n+1/3} |
| 632 |
- \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3}) |
- \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3}) |
| 633 |
+ \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\ |
+ \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\ |
| 634 |
\tau^{n+3/3} & = & \tau^{n} |
\tau^{n+3/3} & = & \tau^{n+2/3} |
| 635 |
- \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3}) |
- \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3}) |
| 636 |
+ \tau^{n} \frac{1}{\Delta r} \delta_i w \right) |
+ \tau^{n} \frac{1}{\Delta r} \delta_i w \right) |
| 637 |
\end{eqnarray} |
\end{eqnarray} |
| 665 |
|
|
| 666 |
\end{minipage} } |
\end{minipage} } |
| 667 |
|
|
| 668 |
|
\begin{figure} |
| 669 |
|
\resizebox{3.5in}{!}{\includegraphics{part2/multiDim_CS.eps}} |
| 670 |
|
\caption{Muti-dimensional advection time-stepping with Cubed-Sphere topology |
| 671 |
|
\label{fig:advect-multidim_cs} |
| 672 |
|
} |
| 673 |
|
\end{figure} |
| 674 |
|
|
| 675 |
\section{Comparison of advection schemes} |
\section{Comparison of advection schemes} |
| 676 |
|
\label{sect:tracer_advection_schemes} |
| 677 |
\begin{rawhtml} |
\begin{rawhtml} |
| 678 |
<!-- CMIREDIR:comparison_of_advection_schemes: --> |
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| 679 |
\end{rawhtml} |
\end{rawhtml} |
| 685 |
Advection Scheme & code & use & use Multi- & Stencil & comments \\ |
Advection Scheme & code & use & use Multi- & Stencil & comments \\ |
| 686 |
& & A.B. & dimension & (1 dim) & \\ |
& & A.B. & dimension & (1 dim) & \\ |
| 687 |
\hline \hline |
\hline \hline |
| 688 |
|
$1^{rst}$order upwind & 1 & No & Yes & 3 pts & linear/$\tau$, non-linear/v\\ |
| 689 |
|
\hline |
| 690 |
centered $2^{nd}$order & 2 & Yes & No & 3 pts & linear \\ |
centered $2^{nd}$order & 2 & Yes & No & 3 pts & linear \\ |
| 691 |
\hline |
\hline |
| 692 |
$3^{rd}$order upwind & 3 & Yes & No & 5 pts & linear/$\tau$\\ |
$3^{rd}$order upwind & 3 & Yes & No & 5 pts & linear/$\tau$\\ |
| 693 |
\hline |
\hline |
| 694 |
centered $4^{th}$order & 4 & Yes & No & 5 pts & linear \\ |
centered $4^{th}$order & 4 & Yes & No & 5 pts & linear \\ |
| 695 |
\hline \hline |
\hline \hline |
| 696 |
% Lax-Wendroff & 10 & No & Yes & 3 pts & linear/tracer, non-linear/flow\\ |
$2^{nd}$order DST (Lax-Wendroff) & 20 & |
| 697 |
% \hline |
No & Yes & 3 pts & linear/$\tau$, non-linear/v\\ |
| 698 |
|
\hline |
| 699 |
$3^{rd}$order DST & 30 & No & Yes & 5 pts & linear/$\tau$, non-linear/v\\ |
$3^{rd}$order DST & 30 & No & Yes & 5 pts & linear/$\tau$, non-linear/v\\ |
| 700 |
\hline \hline |
\hline \hline |
| 701 |
$2^{nd}$order Flux Limiters & 77 & No & Yes & 5 pts & non-linear \\ |
$2^{nd}$order Flux Limiters & 77 & No & Yes & 5 pts & non-linear \\ |
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