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-revision 1.3 by adcroft,
Tue Sep 25 20:13:42 2001 UTC
+revision 1.22 by jmc,
Wed Jun 28 17:01:34 2006 UTC
 Line 2
  % $Name$
  \section{Tracer equations}
+ \label{sect:tracer_equations}
+ \begin{rawhtml}
+ <!-- CMIREDIR:tracer_equations: -->
+ \end{rawhtml}
  The basic discretization used for the tracer equations is the second
  order piece-wise constant finite volume form of the forced
- advection-diussion equations. There are many alternatives to second
+ advection-diffusion equations. There are many alternatives to second
  order method for advection and alternative parameterizations for the
  sub-grid scale processes. The Gent-McWilliams eddy parameterization,
  KPP mixing scheme and PV flux parameterization are all dealt with in
-Line 14 
 part of the tracer equations and the var
+Line 18 
 part of the tracer equations and the var
  described here.
  \subsection{Time-stepping of tracers: ABII}
+ \label{sect:tracer_equations_abII}
+ \begin{rawhtml}
+ <!-- CMIREDIR:tracer_equations_abII: -->
+ \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
-Line 41 
 only affects the surface layer since the
+Line 49 
 only affects the surface layer since the
  everywhere else. This term is therefore referred to as the surface
  correction term. Global conservation is not possible using the
  flux-form (as here) and a linearized free-surface
- (\cite{Griffies00,Campin02}).
+ (\cite{griffies:00,campin:02}).
  The continuity equation can be recovered by setting
  $G_{diff}=G_{forc}=0$ and $\tau=1$.
- The driver routine that calls the routines to calculate tendancies are
+ The driver routine that calls the routines to calculate tendencies are
  {\em S/R CALC\_GT} and {\em S/R CALC\_GS} for temperature and salt
  (moisture), respectively. These in turn call a generic advection
  diffusion routine {\em S/R GAD\_CALC\_RHS} that is called with the
- flow field and relevent tracer as arguments and returns the collective
+ flow field and relevant tracer as arguments and returns the collective
- tendancy due to advection and diffusion. Forcing is add subsequently
+ tendency due to advection and diffusion. Forcing is add subsequently
- in {\em S/R CALC\_GT} or {\em S/R CALC\_GS} to the same tendancy
+ in {\em S/R CALC\_GT} or {\em S/R CALC\_GS} to the same tendency
  array.
  \fbox{ \begin{minipage}{4.75in}
-Line 66 
 $F_r$: {\bf fVerT} (argument)
+Line 74 
 $F_r$: {\bf fVerT} (argument)
  \end{minipage} }
+ The space and time discretization are treated separately (method of
- The space and time discretizations are treated seperately (method of
+ lines). Tendencies are calculated at time levels $n$ and $n-1$ and
- lines). The Adams-Bashforth time discretization reads:
+ 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 tendency at $n-1$ is not re-calculated but rather the
+ tendency at $n$ is stored in a global array for later re-use.
- Strictly speaking the ABII scheme should be applied only to the
+ \fbox{ \begin{minipage}{4.75in}
- advection terms. However, this scheme is only used in conjuction with
+ {\em S/R ADAMS\_BASHFORTH2} ({\em model/src/adams\_bashforth2.F})
- the standard second, third and fourth order advection
- schemes. Selection of any other advection scheme disables
+ $G^{(n+1/2)}$: {\bf gTracer} (argument on exit)
- Adams-Bashforth for tracers so that explicit diffusion and forcing use
- the forward method.
+ $G^{(n)}$: {\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 tendency:
+ \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 conjunction 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}
+ \label{sect:tracer-advection}
+ \begin{rawhtml}
+ <!-- CMIREDIR:linear_advection_schemes: -->
+ \end{rawhtml}
  \begin{figure}
  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}
  \caption{
-Line 136 
 $\mu=c/(1-c)$.
+Line 171 
 $\mu=c/(1-c)$.
  }
  \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
-Line 148 
 commonly used in the field and most fami
+Line 181 
 commonly used in the field and most fami
  \subsection{Centered second order advection-diffusion}
  The basic discretization, centered second order, is the default. It is
- designed to be consistant with the continuity equation to facilitate
+ designed to be consistent with the continuity equation to facilitate
  conservation properties analogous to the continuum. However, centered
- second order advection is notoriously noisey and must be used in
+ second order advection is notoriously noisy and must be used in
- conjuction with some finite amount of diffusion to produce a sensible
+ conjunction with some finite amount of diffusion to produce a sensible
  solution.
  The advection operator is discretized:
-Line 177 
 W & = & {\cal A}_c w
+Line 210 
 W & = & {\cal A}_c w
  For non-divergent flow, this discretization can be shown to conserve
  the tracer both locally and globally and to globally conserve tracer
- variance, $\tau^2$. The proof is given in \cite{Adcroft95,Adcroft97}.
+ variance, $\tau^2$. The proof is given in \cite{adcroft:95,adcroft:97}.
  \fbox{ \begin{minipage}{4.75in}
  {\em S/R GAD\_C2\_ADV\_X} ({\em gad\_c2\_adv\_x.F})
-Line 225 
 F_r & = & W \overline{\tau - \frac{1}{6}
+Line 258 
 F_r & = & W \overline{\tau - \frac{1}{6}
  \end{eqnarray}
  At boundaries, $\delta_{\hat{n}} \tau$ is set to zero allowing
- $\delta_{nn}$ to be evaluated. We are currently examing the accuracy
+ $\delta_{nn}$ to be evaluated. We are currently examine 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}
  Centered fourth order advection is formally the most accurate scheme
  we have implemented and can be used to great effect in high resolution
- simultation where dynamical scales are well resolved. However, the
+ simulation where dynamical scales are well resolved. However, the
- scheme is noisey like the centered second order method and so must be
+ scheme is noisy like the centered second order method and so must be
  used with some finite amount of diffusion. Bi-harmonic is recommended
  since it is more scale selective and less likely to diffuse away the
  well resolved gradient the fourth order scheme worked so hard to
-Line 248 
 F_r & = & W \overline{\tau - \frac{1}{6}
+Line 307 
 F_r & = & W \overline{\tau - \frac{1}{6}
  \end{eqnarray}
  As for the third order scheme, the best discretization near boundaries
- is under investigation but currenlty $\delta_i \tau=0$ on a boundary.
+ is under investigation but currently $\delta_i \tau=0$ 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}
-Line 272 
 if the limiter is set to zero.
+Line 359 
 if the limiter is set to zero.
  \section{Non-linear advection schemes}
+ \begin{rawhtml}
+ <!-- CMIREDIR:non-linear_advection_schemes: -->
+ \end{rawhtml}
  Non-linear advection schemes invoke non-linear interpolation and are
  widely used in computational fluid dynamics (non-linear does not refer
-Line 310 
 r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \t
+Line 400 
 r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \t
  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}
+ \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}
  The direct-space-time method deals with space and time discretization
- together (other methods that treat space and time seperately are known
+ together (other methods that treat space and time separately are known
  collectively as the ``Method of Lines''). The Lax-Wendroff scheme
  falls into this category; it adds sufficient diffusion to a second
  order flux that the forward-in-time method is stable. The upwind
-Line 337 
 where
+Line 454 
 where
  d_1 & = & \frac{1}{6} ( 2 - |c| ) ( 1 - |c| ) \\
  d_2 & = & \frac{1}{6} ( 1 - |c| ) ( 1 + |c| )
  \end{eqnarray}
- The coefficients $d_0$ and $d_1$ approach $1/3$ and $1/6$ repectively
+ The coefficients $d_0$ and $d_1$ approach $1/3$ and $1/6$ respectively
  as the Courant number, $c$, vanishes. In this limit, the conventional
  third order upwind method is recovered. For finite Courant number, the
  deviations from the linear method are analogous to the diffusion added
-Line 345 
 to centered second order advection in th
+Line 462 
 to centered second order advection in th
  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
-Line 353 
 large Courant number, where the linear u
+Line 470 
 large Courant number, where the linear u
  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.
-Line 373 
 and the limiter is the Sweby limiter:
+Line 518 
 and the limiter is the Sweby limiter:
  \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}
- In many of the aforementioned advection schemes the behaviour in
+ \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 Gaussian feature. Courant number is 0.01 with
+$\times$30 points and solutions are shown for T=1/2. White lines
+ indicate zero crossing (ie. the presence of false minima).  The left
+ column shows the second order schemes; top) centered second order with
+ Adams-Bashforth, middle) Lax-Wendroff and bottom) Superbee flux
+ limited. The middle column shows the third order schemes; top) upwind
+ biased third order with Adams-Bashforth, middle) third order direct
+ space-time method and bottom) the same with flux limiting. The top
+ right panel shows the centered fourth order scheme with
+ Adams-Bashforth and right middle panel shows a fourth order variant on
+ the DST method. Bottom right panel shows the Superbee flux limiter
+ (second order) applied independently in each direction (method of
+ lines).
+ \label{fig:advect-2d-lo-diag}
+ }
+ \end{figure}
+ \begin{figure}
+ \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-mid-diag.eps}}
+ \caption{
+ Comparison of advection schemes in two dimensions; diagonal advection
+ of a resolved Gaussian feature. Courant number is 0.27 with
+$\times$30 points and solutions are shown for T=1/2. White lines
+ indicate zero crossing (ie. the presence of false minima).  The left
+ column shows the second order schemes; top) centered second order with
+ Adams-Bashforth, middle) Lax-Wendroff and bottom) Superbee flux
+ limited. The middle column shows the third order schemes; top) upwind
+ biased third order with Adams-Bashforth, middle) third order direct
+ space-time method and bottom) the same with flux limiting. The top
+ right panel shows the centered fourth order scheme with
+ Adams-Bashforth and right middle panel shows a fourth order variant on
+ the DST method. Bottom right panel shows the Superbee flux limiter
+ (second order) applied independently in each direction (method of
+ lines).
+ \label{fig:advect-2d-mid-diag}
+ }
+ \end{figure}
+ \begin{figure}
+ \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-hi-diag.eps}}
+ \caption{
+ Comparison of advection schemes in two dimensions; diagonal advection
+ of a resolved Gaussian feature. Courant number is 0.47 with
+$\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).
+ \label{fig:advect-2d-hi-diag}
+ }
+ \end{figure}
+ In many of the aforementioned advection schemes the behavior in
  multiple dimensions is not necessarily as good as the one dimensional
- behaviour. For instance, a shape preserving monotonic scheme in one
+ 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
-Line 389 
 as if in one dimension:
+Line 627 
 as if in one dimension:
  \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}
  In order to incorporate this method into the general model algorithm,
- we compute the effective tendancy rather than update the tracer so
+ 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:
  \begin{equation}
-Line 408 
 So that the over all time-stepping looks
+Line 646 
 So that the over all time-stepping looks
  \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}
+ \label{sect:tracer_advection_schemes}
+ \begin{rawhtml}
+ <!-- CMIREDIR:comparison_of_advection_schemes: -->
+ \end{rawhtml}
+ \begin{table}[htb]
+ \centering
+  \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 \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
+  \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
+ 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.~\ref{fig:advect-1d-lo} and \ref{fig:advect-1d-hi} 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;
+ \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 propagating 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 stability (up to Courant number 1).
+ \item 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.
+ \item The presence of false extrema is non-physical and this alone is the
+ strongest argument for using a positive scheme.
+ \end{itemize}

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