--- manual/s_algorithm/text/tracer.tex	2001/09/25 20:13:42	1.3
+++ manual/s_algorithm/text/tracer.tex	2001/10/24 23:14:44	1.7
@@ -1,7 +1,8 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/tracer.tex,v 1.3 2001/09/25 20:13:42 adcroft Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/tracer.tex,v 1.7 2001/10/24 23:14:44 cnh Exp $
 % $Name:  $
 
 \section{Tracer equations}
+\label{sec:tracer_equations}
 
 The basic discretization used for the tracer equations is the second
 order piece-wise constant finite volume form of the forced
@@ -14,6 +15,7 @@
 described here.
 
 \subsection{Time-stepping of tracers: ABII}
+\label{sec:tracer_equations_abII}
 
 The default advection scheme is the centered second order method which
 requires a second order or quasi-second order time-stepping scheme to
@@ -66,39 +68,62 @@
 
 \end{minipage} }
 
-
 The space and time discretizations are treated seperately (method of
-lines). The Adams-Bashforth time discretization reads:
+lines). Tendancies are calculated at time levels $n$ and $n-1$ and
+extrapolated to $n+1/2$ using the Adams-Bashforth method:
 \marginpar{$\epsilon$: {\bf AB\_eps}}
-\marginpar{$\Delta t$: {\bf deltaTtracer}}
 \begin{equation}
-\tau^{(n+1)} = \tau^{(n)} + \Delta t \left(
+G^{(n+1/2)} = 
 (\frac{3}{2} + \epsilon) G^{(n)} - (\frac{1}{2} + \epsilon) G^{(n-1)}
-\right)
 \end{equation}
 where $G^{(n)} = G_{adv}^\tau + G_{diff}^\tau + G_{src}^\tau$ at time
-step $n$.
+step $n$. The tendancy at $n-1$ is not re-calculated but rather the
+tendancy at $n$ is stored in a global array for later re-use.
 
-Strictly speaking the ABII scheme should be applied only to the
-advection terms. However, this scheme is only used in conjuction with
-the standard second, third and fourth order advection
-schemes. Selection of any other advection scheme disables
-Adams-Bashforth for tracers so that explicit diffusion and forcing use
-the forward method.
+\fbox{ \begin{minipage}{4.75in}
+{\em S/R ADAMS\_BASHFORTH2} ({\em model/src/adams\_bashforth2.F})
+
+$G^{(n+1/2)}$: {\bf gTracer} (argument on exit)
+
+$G^{(n)}$: {\bf gTracer} (argument on entry)
+
+$G^{(n-1)}$: {\bf gTrNm1} (argument)
+
+$\epsilon$: {\bf ABeps} (PARAMS.h)
+
+\end{minipage} }
+
+The tracers are stepped forward in time using the extrapolated tendancy:
+\begin{equation}
+\tau^{(n+1)} = \tau^{(n)} + \Delta t G^{(n+1/2)}
+\end{equation}
+\marginpar{$\Delta t$: {\bf deltaTtracer}}
 
 \fbox{ \begin{minipage}{4.75in}
 {\em S/R TIMESTEP\_TRACER} ({\em model/src/timestep\_tracer.F})
 
-$\tau$: {\bf tracer} (argument)
+$\tau^{(n+1)}$: {\bf gTracer} (argument on exit)
 
-$G^{(n)}$: {\bf gTracer} (argument)
+$\tau^{(n)}$: {\bf tracer} (argument on entry)
 
-$G^{(n-1)}$: {\bf gTrNm1} (argument)
+$G^{(n+1/2)}$: {\bf gTracer} (argument)
 
 $\Delta t$: {\bf deltaTtracer} (PARAMS.h)
 
 \end{minipage} }
 
+Strictly speaking the ABII scheme should be applied only to the
+advection terms. However, this scheme is only used in conjuction with
+the standard second, third and fourth order advection
+schemes. Selection of any other advection scheme disables
+Adams-Bashforth for tracers so that explicit diffusion and forcing use
+the forward method.
+
+
+
+
+\section{Linear advection schemes}
+
 \begin{figure}
 \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}
 \caption{
@@ -136,8 +161,6 @@
 }
 \end{figure}
 
-\section{Linear advection schemes}
-
 The advection schemes known as centered second order, centered fourth
 order, first order upwind and upwind biased third order are known as
 linear advection schemes because the coefficient for interpolation of
@@ -228,6 +251,32 @@
 $\delta_{nn}$ to be evaluated. We are currently examing the accuracy
 of this boundary condition and the effect on the solution.
 
+\fbox{ \begin{minipage}{4.75in}
+{\em S/R GAD\_U3\_ADV\_X} ({\em gad\_u3\_adv\_x.F})
+
+$F_x$: {\bf uT} (argument)
+
+$U$: {\bf uTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+{\em S/R GAD\_U3\_ADV\_Y} ({\em gad\_u3\_adv\_y.F})
+
+$F_y$: {\bf vT} (argument)
+
+$V$: {\bf vTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+{\em S/R GAD\_U3\_ADV\_R} ({\em gad\_u3\_adv\_r.F})
+
+$F_r$: {\bf wT} (argument)
+
+$W$: {\bf rTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+\end{minipage} }
 
 \subsection{Centered fourth order advection}
 
@@ -250,6 +299,34 @@
 As for the third order scheme, the best discretization near boundaries
 is under investigation but currenlty $\delta_i \tau=0$ on a boundary.
 
+\fbox{ \begin{minipage}{4.75in}
+{\em S/R GAD\_C4\_ADV\_X} ({\em gad\_c4\_adv\_x.F})
+
+$F_x$: {\bf uT} (argument)
+
+$U$: {\bf uTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+{\em S/R GAD\_C4\_ADV\_Y} ({\em gad\_c4\_adv\_y.F})
+
+$F_y$: {\bf vT} (argument)
+
+$V$: {\bf vTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+{\em S/R GAD\_C4\_ADV\_R} ({\em gad\_c4\_adv\_r.F})
+
+$F_r$: {\bf wT} (argument)
+
+$W$: {\bf rTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+\end{minipage} }
+
+
 \subsection{First order upwind advection}
 
 Although the upwind scheme is the underlying scheme for the robust or
@@ -315,6 +392,33 @@
 \psi(r) = \max[0,\min[1,2r],\min[2,r]]
 \end{equation}
 
+\fbox{ \begin{minipage}{4.75in}
+{\em S/R GAD\_FLUXLIMIT\_ADV\_X} ({\em gad\_fluxlimit\_adv\_x.F})
+
+$F_x$: {\bf uT} (argument)
+
+$U$: {\bf uTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+{\em S/R GAD\_FLUXLIMIT\_ADV\_Y} ({\em gad\_fluxlimit\_adv\_y.F})
+
+$F_y$: {\bf vT} (argument)
+
+$V$: {\bf vTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+{\em S/R GAD\_FLUXLIMIT\_ADV\_R} ({\em gad\_fluxlimit\_adv\_r.F})
+
+$F_r$: {\bf wT} (argument)
+
+$W$: {\bf rTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+\end{minipage} }
+
 
 \subsection{Third order direct space time}
 
@@ -353,6 +457,34 @@
 unstable, the scheme is extremely accurate
 (Fig.~\ref{fig:advect-1d-hi}) with only minor overshoots.
 
+\fbox{ \begin{minipage}{4.75in}
+{\em S/R GAD\_DST3\_ADV\_X} ({\em gad\_dst3\_adv\_x.F})
+
+$F_x$: {\bf uT} (argument)
+
+$U$: {\bf uTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+{\em S/R GAD\_DST3\_ADV\_Y} ({\em gad\_dst3\_adv\_y.F})
+
+$F_y$: {\bf vT} (argument)
+
+$V$: {\bf vTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+{\em S/R GAD\_DST3\_ADV\_R} ({\em gad\_dst3\_adv\_r.F})
+
+$F_r$: {\bf wT} (argument)
+
+$W$: {\bf rTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+\end{minipage} }
+
+
 \subsection{Third order direct space time with flux limiting}
 
 The overshoots in the DST3 method can be controlled with a flux limiter.
@@ -373,8 +505,101 @@
 \psi(r) = \max[0, \min[\min(1,d_0+d_1r],\frac{1-c}{c}r ]]
 \end{equation}
 
+\fbox{ \begin{minipage}{4.75in}
+{\em S/R GAD\_DST3FL\_ADV\_X} ({\em gad\_dst3\_adv\_x.F})
+
+$F_x$: {\bf uT} (argument)
+
+$U$: {\bf uTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+{\em S/R GAD\_DST3FL\_ADV\_Y} ({\em gad\_dst3\_adv\_y.F})
+
+$F_y$: {\bf vT} (argument)
+
+$V$: {\bf vTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+{\em S/R GAD\_DST3FL\_ADV\_R} ({\em gad\_dst3\_adv\_r.F})
+
+$F_r$: {\bf wT} (argument)
+
+$W$: {\bf rTrans} (argument)
+
+$\tau$: {\bf tracer} (argument)
+
+\end{minipage} }
+
+
 \subsection{Multi-dimensional advection}
 
+\begin{figure}
+\resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-lo-diag.eps}}
+\caption{
+Comparison of advection schemes in two dimensions; diagonal advection
+of a resolved Guassian feature. Courant number is 0.01 with
+30$\times$30 points and solutions are shown for T=1/2. White lines
+indicate zero crossing (ie. the presence of false minima).  The left
+column shows the second order schemes; top) centered second order with
+Adams-Bashforth, middle) Lax-Wendroff and bottom) Superbee flux
+limited. The middle column shows the third order schemes; top) upwind
+biased third order with Adams-Bashforth, middle) third order direct
+space-time method and bottom) the same with flux limiting. The top
+right panel shows the centered fourth order scheme with
+Adams-Bashforth and right middle panel shows a fourth order variant on
+the DST method. Bottom right panel shows the Superbee flux limiter
+(second order) applied independantly in each direction (method of
+lines).
+\label{fig:advect-2d-lo-diag}
+}
+\end{figure}
+
+\begin{figure}
+\resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-mid-diag.eps}}
+\caption{
+Comparison of advection schemes in two dimensions; diagonal advection
+of a resolved Guassian feature. Courant number is 0.27 with
+30$\times$30 points and solutions are shown for T=1/2. White lines
+indicate zero crossing (ie. the presence of false minima).  The left
+column shows the second order schemes; top) centered second order with
+Adams-Bashforth, middle) Lax-Wendroff and bottom) Superbee flux
+limited. The middle column shows the third order schemes; top) upwind
+biased third order with Adams-Bashforth, middle) third order direct
+space-time method and bottom) the same with flux limiting. The top
+right panel shows the centered fourth order scheme with
+Adams-Bashforth and right middle panel shows a fourth order variant on
+the DST method. Bottom right panel shows the Superbee flux limiter
+(second order) applied independantly in each direction (method of
+lines).
+\label{fig:advect-2d-mid-diag}
+}
+\end{figure}
+
+\begin{figure}
+\resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-hi-diag.eps}}
+\caption{
+Comparison of advection schemes in two dimensions; diagonal advection
+of a resolved Guassian feature. Courant number is 0.47 with
+30$\times$30 points and solutions are shown for T=1/2. White lines
+indicate zero crossings and initial maximum values (ie. the presence
+of false extrema).  The left column shows the second order schemes;
+top) centered second order with Adams-Bashforth, middle) Lax-Wendroff
+and bottom) Superbee flux limited. The middle column shows the third
+order schemes; top) upwind biased third order with Adams-Bashforth,
+middle) third order direct space-time method and bottom) the same with
+flux limiting. The top right panel shows the centered fourth order
+scheme with Adams-Bashforth and right middle panel shows a fourth
+order variant on the DST method. Bottom right panel shows the Superbee
+flux limiter (second order) applied independantly in each direction
+(method of lines).
+\label{fig:advect-2d-hi-diag}
+}
+\end{figure}
+
+
+
 In many of the aforementioned advection schemes the behaviour in
 multiple dimensions is not necessarily as good as the one dimensional
 behaviour. For instance, a shape preserving monotonic scheme in one
@@ -408,3 +633,77 @@
 \begin{equation}
 \tau^{n+1} = \tau^{n} + \Delta t \left( G^{n+1/2}_{adv} + G_{diff}(\tau^{n}) + G^{n}_{forcing} \right)
 \end{equation}
+
+\fbox{ \begin{minipage}{4.75in}
+{\em S/R GAD\_ADVECTION} ({\em gad\_advection.F})
+
+$\tau$: {\bf Tracer} (argument)
+
+$G^{n+1/2}_{adv}$: {\bf Gtracer} (argument)
+
+$F_x, F_y, F_r$: {\bf af} (local)
+
+$U$: {\bf uTrans} (local)
+
+$V$: {\bf vTrans} (local)
+
+$W$: {\bf rTrans} (local)
+
+\end{minipage} }
+
+
+\section{Comparison of advection schemes}
+
+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
+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 tendancy is to loose amplitude in sharp
+peaks due to diffusion. The one dimensional tests shown in
+Figs.~\ref{fig:advect-1d-lo} and \ref{fig:advect-1d-hi} showed this
+phenomenum.
+
+Finally, the bottom left and right panels use the same advection
+scheme but the right does not use the mutli-dimensional method. At low
+Courant number this appears to not matter but for moderate Courant
+number severe distortion of the feature is apparent. Moreoever, 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;
+\begin{itemize}
+\item 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.
+\item 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.
+\item If your solution has shocks or propagatin fronts then a
+flux limited scheme is almost essential.
+\item If your time-step is limited by advection, the multi-dimensional
+non-linear schemes have the most stablility (upto Courant number 1).
+\item If you need to know how much diffusion/dissipation has occured you
+will have a lot of trouble figuring it out with a non-linear method.
+\item The presence of false extrema is unphysical and this alone is the
+strongest argument for using a positive scheme.
+\end{itemize}