--- manual/s_algorithm/text/tracer.tex 2001/11/13 18:15:26 1.10 +++ manual/s_algorithm/text/tracer.tex 2016/10/26 17:30:23 1.27 @@ -1,8 +1,11 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/tracer.tex,v 1.10 2001/11/13 18:15:26 adcroft Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/tracer.tex,v 1.27 2016/10/26 17:30:23 heimbach Exp $ % $Name: $ \section{Tracer equations} \label{sec:tracer_equations} +\begin{rawhtml} + +\end{rawhtml} The basic discretization used for the tracer equations is the second order piece-wise constant finite volume form of the forced @@ -16,6 +19,9 @@ \subsection{Time-stepping of tracers: ABII} \label{sec:tracer_equations_abII} +\begin{rawhtml} + +\end{rawhtml} The default advection scheme is the centered second order method which requires a second order or quasi-second order time-stepping scheme to @@ -123,9 +129,13 @@ \section{Linear advection schemes} +\label{sec:tracer-advection} +\begin{rawhtml} + +\end{rawhtml} \begin{figure} -\resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}} +\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-lo.eps}} \caption{ Comparison of 1-D advection schemes. Courant number is 0.05 with 60 points and solutions are shown for T=1 (one complete period). @@ -143,7 +153,7 @@ \end{figure} \begin{figure} -\resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-hi.eps}} +\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-hi.eps}} \caption{ Comparison of 1-D advection schemes. Courant number is 0.89 with 60 points and solutions are shown for T=1 (one complete period). @@ -349,6 +359,10 @@ \section{Non-linear advection schemes} +\label{sec:non-linear_advection_schemes} +\begin{rawhtml} + +\end{rawhtml} Non-linear advection schemes invoke non-linear interpolation and are widely used in computational fluid dynamics (non-linear does not refer @@ -387,7 +401,7 @@ r = \frac{ \tau_{i+1} - \tau_{i} }{ \tau_{i} - \tau_{i-1} } & \forall & u < 0 \end{eqnarray} as it's argument. There are many choices of limiter function but we -only provide the Superbee limiter \cite{Roe85}: +only provide the Superbee limiter \cite{roe:85}: \begin{equation} \psi(r) = \max[0,\min[1,2r],\min[2,r]] \end{equation} @@ -449,7 +463,7 @@ The DST3 method described above must be used in a forward-in-time manner and is stable for $0 \le |c| \le 1$. Although the scheme -appears to be forward-in-time, it is in fact second order in time and +appears to be forward-in-time, it is in fact third order in time and the accuracy increases with the Courant number! For low Courant number, DST3 produces very similar results (indistinguishable in Fig.~\ref{fig:advect-1d-lo}) to the linear third order method but for @@ -536,7 +550,7 @@ \subsection{Multi-dimensional advection} \begin{figure} -\resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-lo-diag.eps}} +\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-lo-diag.eps}} \caption{ Comparison of advection schemes in two dimensions; diagonal advection of a resolved Gaussian feature. Courant number is 0.01 with @@ -557,7 +571,7 @@ \end{figure} \begin{figure} -\resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-mid-diag.eps}} +\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-mid-diag.eps}} \caption{ Comparison of advection schemes in two dimensions; diagonal advection of a resolved Gaussian feature. Courant number is 0.27 with @@ -578,7 +592,7 @@ \end{figure} \begin{figure} -\resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-hi-diag.eps}} +\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-hi-diag.eps}} \caption{ Comparison of advection schemes in two dimensions; diagonal advection of a resolved Gaussian feature. Courant number is 0.47 with @@ -614,10 +628,10 @@ \tau^{n+1/3} & = & \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) \\ -\tau^{n+2/3} & = & \tau^{n} +\tau^{n+2/3} & = & \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) \\ -\tau^{n+3/3} & = & \tau^{n} +\tau^{n+3/3} & = & \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) \end{eqnarray} @@ -651,8 +665,74 @@ \end{minipage} } +\begin{figure} +\resizebox{3.5in}{!}{\includegraphics{s_algorithm/figs/multiDim_CS.eps}} +\caption{Muti-dimensional advection time-stepping with Cubed-Sphere topology +\label{fig:advect-multidim_cs} +} +\end{figure} \section{Comparison of advection schemes} +\label{sec:tracer_advection_schemes} +\begin{rawhtml} + +\end{rawhtml} + +\begin{table}[htb] +\centering +{\small + \begin{tabular}[htb]{|l|c|c|c|c|l|} + \hline + Advection Scheme & code & use & use Multi- & Stencil & comments \\ + & & A.B. & dimension & (1 dim) & \\ + \hline \hline + $1^{rst}$order upwind & 1 & No & Yes & 3 pts & linear/$\tau$, non-linear/v\\ + \hline + centered $2^{nd}$order & 2 & Yes & No & 3 pts & linear \\ + \hline + $3^{rd}$order upwind & 3 & Yes & No & 5 pts & linear/$\tau$\\ + \hline + centered $4^{th}$order & 4 & Yes & No & 5 pts & linear \\ + \hline \hline + $2^{nd}$order DST (Lax-Wendroff) & 20 & + No & Yes & 3 pts & linear/$\tau$, non-linear/v\\ + \hline + $3^{rd}$order DST & 30 & No & Yes & 5 pts & linear/$\tau$, non-linear/v\\ + \hline + $2^{nd}$order-moment Prather & 80 & No & Yes & ~ & ~ \\ + \hline \hline + $2^{nd}$order Flux Limiters & 77 & No & Yes & 5 pts & non-linear \\ + \hline + $3^{nd}$order DST Flux limiter & 33 & No & Yes & 5 pts & non-linear \\ + \hline + $2^{nd}$order-moment Prather w. limiter & 81 & No & Yes & ~ & ~ \\ + \hline + piecewise parabolic w. ``null'' limiter & 40 & No & Yes & ~ & ~ \\ + \hline + piecewise parabolic w. ``mono'' limiter & 41 & No & Yes & ~ & ~ \\ + \hline + piecewise quartic w. ``null'' limiter & 50 & No & Yes & ~ & ~ \\ + \hline + piecewise quartic w. ``mono'' limiter & 51 & No & Yes & ~ & ~ \\ + \hline + piecewise quartic w. ``weno'' limiter & 52 & No & Yes & ~ & ~ \\ + \hline + $7^{nd}$order one-step method & 7 & No & Yes & ~ & ~ \\ + with Monotonicity Preserving Limiter & ~ & ~ & ~ & ~ & ~ \\ + \hline + + \end{tabular} + } + \caption{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., {\bf tempAdvScheme}=3 in + file {\em data} selects the $3^{rd}$ order upwind advection scheme + for temperature). + } + \label{tab:advectionShemes_summary} +\end{table} + Figs.~\ref{fig:advect-2d-lo-diag}, \ref{fig:advect-2d-mid-diag} and \ref{fig:advect-2d-hi-diag} show solutions to a simple diagonal