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% $Header: /u/gcmpack/manual/s_algorithm/text/tracer.tex,v 1.26 2010/08/30 23:09:19 jmc Exp $ |
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% $Name: $ |
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|
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\section{Tracer equations} |
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\label{sec:tracer_equations} |
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\begin{rawhtml} |
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<!-- CMIREDIR:tracer_equations: --> |
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\end{rawhtml} |
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|
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The basic discretization used for the tracer equations is the second |
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order piece-wise constant finite volume form of the forced |
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advection-diffusion equations. There are many alternatives to second |
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order method for advection and alternative parameterizations for the |
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sub-grid scale processes. The Gent-McWilliams eddy parameterization, |
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KPP mixing scheme and PV flux parameterization are all dealt with in |
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separate sections. The basic discretization of the advection-diffusion |
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part of the tracer equations and the various advection schemes will be |
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described here. |
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|
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\subsection{Time-stepping of tracers: ABII} |
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\label{sec:tracer_equations_abII} |
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\begin{rawhtml} |
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<!-- CMIREDIR:tracer_equations_abII: --> |
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\end{rawhtml} |
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|
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The default advection scheme is the centered second order method which |
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requires a second order or quasi-second order time-stepping scheme to |
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be stable. Historically this has been the quasi-second order |
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Adams-Bashforth method (ABII) and applied to all terms. For an |
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arbitrary tracer, $\tau$, the forced advection-diffusion equation |
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reads: |
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\begin{equation} |
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\partial_t \tau + G_{adv}^\tau = G_{diff}^\tau + G_{forc}^\tau |
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\end{equation} |
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where $G_{adv}^\tau$, $G_{diff}^\tau$ and $G_{forc}^\tau$ are the |
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tendencies due to advection, diffusion and forcing, respectively, |
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namely: |
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\begin{eqnarray} |
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G_{adv}^\tau & = & \partial_x u \tau + \partial_y v \tau + \partial_r w \tau |
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- \tau \nabla \cdot {\bf v} \\ |
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G_{diff}^\tau & = & \nabla \cdot {\bf K} \nabla \tau |
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\end{eqnarray} |
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and the forcing can be some arbitrary function of state, time and |
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space. |
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|
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The term, $\tau \nabla \cdot {\bf v}$, is required to retain local |
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conservation in conjunction with the linear implicit free-surface. It |
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only affects the surface layer since the flow is non-divergent |
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everywhere else. This term is therefore referred to as the surface |
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correction term. Global conservation is not possible using the |
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flux-form (as here) and a linearized free-surface |
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(\cite{griffies:00,campin:02}). |
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|
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The continuity equation can be recovered by setting |
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$G_{diff}=G_{forc}=0$ and $\tau=1$. |
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|
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The driver routine that calls the routines to calculate tendencies are |
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{\em S/R CALC\_GT} and {\em S/R CALC\_GS} for temperature and salt |
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(moisture), respectively. These in turn call a generic advection |
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diffusion routine {\em S/R GAD\_CALC\_RHS} that is called with the |
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flow field and relevant tracer as arguments and returns the collective |
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tendency due to advection and diffusion. Forcing is add subsequently |
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in {\em S/R CALC\_GT} or {\em S/R CALC\_GS} to the same tendency |
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array. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R GAD\_CALC\_RHS} ({\em pkg/generic\_advdiff/gad\_calc\_rhs.F}) |
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|
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$\tau$: {\bf tracer} (argument) |
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|
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$G^{(n)}$: {\bf gTracer} (argument) |
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|
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$F_r$: {\bf fVerT} (argument) |
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|
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\end{minipage} } |
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|
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The space and time discretization are treated separately (method of |
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lines). Tendencies are calculated at time levels $n$ and $n-1$ and |
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extrapolated to $n+1/2$ using the Adams-Bashforth method: |
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\marginpar{$\epsilon$: {\bf AB\_eps}} |
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\begin{equation} |
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G^{(n+1/2)} = |
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(\frac{3}{2} + \epsilon) G^{(n)} - (\frac{1}{2} + \epsilon) G^{(n-1)} |
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\end{equation} |
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where $G^{(n)} = G_{adv}^\tau + G_{diff}^\tau + G_{src}^\tau$ at time |
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step $n$. The tendency at $n-1$ is not re-calculated but rather the |
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tendency at $n$ is stored in a global array for later re-use. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R ADAMS\_BASHFORTH2} ({\em model/src/adams\_bashforth2.F}) |
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|
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$G^{(n+1/2)}$: {\bf gTracer} (argument on exit) |
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|
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$G^{(n)}$: {\bf gTracer} (argument on entry) |
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|
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$G^{(n-1)}$: {\bf gTrNm1} (argument) |
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|
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$\epsilon$: {\bf ABeps} (PARAMS.h) |
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|
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\end{minipage} } |
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|
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The tracers are stepped forward in time using the extrapolated tendency: |
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\begin{equation} |
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\tau^{(n+1)} = \tau^{(n)} + \Delta t G^{(n+1/2)} |
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\end{equation} |
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\marginpar{$\Delta t$: {\bf deltaTtracer}} |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R TIMESTEP\_TRACER} ({\em model/src/timestep\_tracer.F}) |
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|
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$\tau^{(n+1)}$: {\bf gTracer} (argument on exit) |
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|
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$\tau^{(n)}$: {\bf tracer} (argument on entry) |
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|
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$G^{(n+1/2)}$: {\bf gTracer} (argument) |
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|
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$\Delta t$: {\bf deltaTtracer} (PARAMS.h) |
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|
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\end{minipage} } |
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|
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Strictly speaking the ABII scheme should be applied only to the |
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advection terms. However, this scheme is only used in conjunction with |
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the standard second, third and fourth order advection |
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schemes. Selection of any other advection scheme disables |
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Adams-Bashforth for tracers so that explicit diffusion and forcing use |
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the forward method. |
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|
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|
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|
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|
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\section{Linear advection schemes} |
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\label{sec:tracer-advection} |
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\begin{rawhtml} |
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<!-- CMIREDIR:linear_advection_schemes: --> |
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\end{rawhtml} |
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|
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\begin{figure} |
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\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-lo.eps}} |
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\caption{ |
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Comparison of 1-D advection schemes. Courant number is 0.05 with 60 |
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points and solutions are shown for T=1 (one complete period). |
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a) Shows the upwind biased schemes; first order upwind, DST3, |
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third order upwind and second order upwind. |
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b) Shows the centered schemes; Lax-Wendroff, DST4, centered second order, |
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centered fourth order and finite volume fourth order. |
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c) Shows the second order flux limiters: minmod, Superbee, |
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MC limiter and the van Leer limiter. |
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d) Shows the DST3 method with flux limiters due to Sweby with |
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$\mu=1$, $\mu=c/(1-c)$ and a fourth order DST method with Sweby limiter, |
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$\mu=c/(1-c)$. |
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\label{fig:advect-1d-lo} |
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} |
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\end{figure} |
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|
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\begin{figure} |
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\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-hi.eps}} |
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\caption{ |
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Comparison of 1-D advection schemes. Courant number is 0.89 with 60 |
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points and solutions are shown for T=1 (one complete period). |
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a) Shows the upwind biased schemes; first order upwind and DST3. |
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Third order upwind and second order upwind are unstable at this Courant number. |
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b) Shows the centered schemes; Lax-Wendroff, DST4. Centered second order, |
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centered fourth order and finite volume fourth order and unstable at this |
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Courant number. |
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c) Shows the second order flux limiters: minmod, Superbee, |
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MC limiter and the van Leer limiter. |
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d) Shows the DST3 method with flux limiters due to Sweby with |
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$\mu=1$, $\mu=c/(1-c)$ and a fourth order DST method with Sweby limiter, |
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$\mu=c/(1-c)$. |
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\label{fig:advect-1d-hi} |
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} |
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\end{figure} |
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|
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The advection schemes known as centered second order, centered fourth |
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order, first order upwind and upwind biased third order are known as |
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linear advection schemes because the coefficient for interpolation of |
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the advected tracer are linear and a function only of the flow, not |
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the tracer field it self. We discuss these first since they are most |
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commonly used in the field and most familiar. |
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|
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\subsection{Centered second order advection-diffusion} |
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|
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The basic discretization, centered second order, is the default. It is |
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designed to be consistent with the continuity equation to facilitate |
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conservation properties analogous to the continuum. However, centered |
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second order advection is notoriously noisy and must be used in |
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conjunction with some finite amount of diffusion to produce a sensible |
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solution. |
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|
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The advection operator is discretized: |
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\begin{equation} |
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{\cal A}_c \Delta r_f h_c G_{adv}^\tau = |
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\delta_i F_x + \delta_j F_y + \delta_k F_r |
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\end{equation} |
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where the area integrated fluxes are given by: |
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\begin{eqnarray} |
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F_x & = & U \overline{ \tau }^i \\ |
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F_y & = & V \overline{ \tau }^j \\ |
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F_r & = & W \overline{ \tau }^k |
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\end{eqnarray} |
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The quantities $U$, $V$ and $W$ are volume fluxes defined: |
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\marginpar{$U$: {\bf uTrans} } |
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\marginpar{$V$: {\bf vTrans} } |
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\marginpar{$W$: {\bf rTrans} } |
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\begin{eqnarray} |
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U & = & \Delta y_g \Delta r_f h_w u \\ |
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V & = & \Delta x_g \Delta r_f h_s v \\ |
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W & = & {\cal A}_c w |
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\end{eqnarray} |
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|
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For non-divergent flow, this discretization can be shown to conserve |
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the tracer both locally and globally and to globally conserve tracer |
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variance, $\tau^2$. The proof is given in \cite{adcroft:95,adcroft:97}. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R GAD\_C2\_ADV\_X} ({\em gad\_c2\_adv\_x.F}) |
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|
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$F_x$: {\bf uT} (argument) |
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|
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$U$: {\bf uTrans} (argument) |
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|
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$\tau$: {\bf tracer} (argument) |
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|
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{\em S/R GAD\_C2\_ADV\_Y} ({\em gad\_c2\_adv\_y.F}) |
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|
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$F_y$: {\bf vT} (argument) |
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|
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$V$: {\bf vTrans} (argument) |
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|
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$\tau$: {\bf tracer} (argument) |
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|
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{\em S/R GAD\_C2\_ADV\_R} ({\em gad\_c2\_adv\_r.F}) |
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|
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$F_r$: {\bf wT} (argument) |
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|
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$W$: {\bf rTrans} (argument) |
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|
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$\tau$: {\bf tracer} (argument) |
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|
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\end{minipage} } |
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|
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|
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\subsection{Third order upwind bias advection} |
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|
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Upwind biased third order advection offers a relatively good |
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compromise between accuracy and smoothness. It is not a ``positive'' |
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scheme meaning false extrema are permitted but the amplitude of such |
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are significantly reduced over the centered second order method. |
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|
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The third order upwind fluxes are discretized: |
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\begin{eqnarray} |
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F_x & = & U \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^i |
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+ \frac{1}{2} |U| \delta_i \frac{1}{6} \delta_{ii} \tau \\ |
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F_y & = & V \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^j |
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+ \frac{1}{2} |V| \delta_j \frac{1}{6} \delta_{jj} \tau \\ |
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F_r & = & W \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^k |
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+ \frac{1}{2} |W| \delta_k \frac{1}{6} \delta_{kk} \tau |
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\end{eqnarray} |
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|
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At boundaries, $\delta_{\hat{n}} \tau$ is set to zero allowing |
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$\delta_{nn}$ to be evaluated. We are currently examine the accuracy |
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of this boundary condition and the effect on the solution. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R GAD\_U3\_ADV\_X} ({\em gad\_u3\_adv\_x.F}) |
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|
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$F_x$: {\bf uT} (argument) |
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|
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$U$: {\bf uTrans} (argument) |
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|
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$\tau$: {\bf tracer} (argument) |
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|
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{\em S/R GAD\_U3\_ADV\_Y} ({\em gad\_u3\_adv\_y.F}) |
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|
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$F_y$: {\bf vT} (argument) |
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|
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$V$: {\bf vTrans} (argument) |
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|
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$\tau$: {\bf tracer} (argument) |
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|
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{\em S/R GAD\_U3\_ADV\_R} ({\em gad\_u3\_adv\_r.F}) |
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|
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$F_r$: {\bf wT} (argument) |
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|
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$W$: {\bf rTrans} (argument) |
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|
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$\tau$: {\bf tracer} (argument) |
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|
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\end{minipage} } |
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|
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\subsection{Centered fourth order advection} |
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|
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Centered fourth order advection is formally the most accurate scheme |
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we have implemented and can be used to great effect in high resolution |
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simulation where dynamical scales are well resolved. However, the |
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scheme is noisy like the centered second order method and so must be |
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used with some finite amount of diffusion. Bi-harmonic is recommended |
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since it is more scale selective and less likely to diffuse away the |
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well resolved gradient the fourth order scheme worked so hard to |
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create. |
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|
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The centered fourth order fluxes are discretized: |
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\begin{eqnarray} |
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F_x & = & U \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^i \\ |
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F_y & = & V \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^j \\ |
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F_r & = & W \overline{\tau - \frac{1}{6} \delta_{ii} \tau}^k |
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\end{eqnarray} |
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|
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As for the third order scheme, the best discretization near boundaries |
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is under investigation but currently $\delta_i \tau=0$ on a boundary. |
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|
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R GAD\_C4\_ADV\_X} ({\em gad\_c4\_adv\_x.F}) |
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|
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$F_x$: {\bf uT} (argument) |
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|
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$U$: {\bf uTrans} (argument) |
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|
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$\tau$: {\bf tracer} (argument) |
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|
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{\em S/R GAD\_C4\_ADV\_Y} ({\em gad\_c4\_adv\_y.F}) |
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|
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$F_y$: {\bf vT} (argument) |
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|
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$V$: {\bf vTrans} (argument) |
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|
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$\tau$: {\bf tracer} (argument) |
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|
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{\em S/R GAD\_C4\_ADV\_R} ({\em gad\_c4\_adv\_r.F}) |
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|
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$F_r$: {\bf wT} (argument) |
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|
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$W$: {\bf rTrans} (argument) |
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|
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$\tau$: {\bf tracer} (argument) |
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|
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\end{minipage} } |
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|
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|
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\subsection{First order upwind advection} |
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|
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Although the upwind scheme is the underlying scheme for the robust or |
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non-linear methods given later, we haven't actually supplied this |
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method for general use. It would be very diffusive and it is unlikely |
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that it could ever produce more useful results than the positive |
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higher order schemes. |
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|
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Upwind bias is introduced into many schemes using the {\em abs} |
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function and is allows the first order upwind flux to be written: |
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\begin{eqnarray} |
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F_x & = & U \overline{ \tau }^i - \frac{1}{2} |U| \delta_i \tau \\ |
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F_y & = & V \overline{ \tau }^j - \frac{1}{2} |V| \delta_j \tau \\ |
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F_r & = & W \overline{ \tau }^k - \frac{1}{2} |W| \delta_k \tau |
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\end{eqnarray} |
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|
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If for some reason, the above method is required, then the second |
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order flux limiter scheme described later reduces to the above scheme |
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if the limiter is set to zero. |
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|
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|
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\section{Non-linear advection schemes} |
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\label{sec:non-linear_advection_schemes} |
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\begin{rawhtml} |
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<!-- CMIREDIR:non-linear_advection_schemes: --> |
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\end{rawhtml} |
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|
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Non-linear advection schemes invoke non-linear interpolation and are |
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widely used in computational fluid dynamics (non-linear does not refer |
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to the non-linearity of the advection operator). The flux limited |
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advection schemes belong to the class of finite volume methods which |
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neatly ties into the spatial discretization of the model. |
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|
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When employing the flux limited schemes, first order upwind or |
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direct-space-time method the time-stepping is switched to forward in |
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time. |
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|
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\subsection{Second order flux limiters} |
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|
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The second order flux limiter method can be cast in several ways but |
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is generally expressed in terms of other flux approximations. For |
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example, in terms of a first order upwind flux and second order |
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Lax-Wendroff flux, the limited flux is given as: |
383 |
\begin{equation} |
384 |
F = F_1 + \psi(r) F_{LW} |
385 |
\end{equation} |
386 |
where $\psi(r)$ is the limiter function, |
387 |
\begin{equation} |
388 |
F_1 = u \overline{\tau}^i - \frac{1}{2} |u| \delta_i \tau |
389 |
\end{equation} |
390 |
is the upwind flux, |
391 |
\begin{equation} |
392 |
F_{LW} = F_1 + \frac{|u|}{2} (1-c) \delta_i \tau |
393 |
\end{equation} |
394 |
is the Lax-Wendroff flux and $c = \frac{u \Delta t}{\Delta x}$ is the |
395 |
Courant (CFL) number. |
396 |
|
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The limiter function, $\psi(r)$, takes the slope ratio |
398 |
\begin{eqnarray} |
399 |
r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \tau_{i} - \tau_{i-1} } & \forall & u > 0 |
400 |
\\ |
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r = \frac{ \tau_{i+1} - \tau_{i} }{ \tau_{i} - \tau_{i-1} } & \forall & u < 0 |
402 |
\end{eqnarray} |
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as it's argument. There are many choices of limiter function but we |
404 |
only provide the Superbee limiter \cite{roe:85}: |
405 |
\begin{equation} |
406 |
\psi(r) = \max[0,\min[1,2r],\min[2,r]] |
407 |
\end{equation} |
408 |
|
409 |
\fbox{ \begin{minipage}{4.75in} |
410 |
{\em S/R GAD\_FLUXLIMIT\_ADV\_X} ({\em gad\_fluxlimit\_adv\_x.F}) |
411 |
|
412 |
$F_x$: {\bf uT} (argument) |
413 |
|
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$U$: {\bf uTrans} (argument) |
415 |
|
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$\tau$: {\bf tracer} (argument) |
417 |
|
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{\em S/R GAD\_FLUXLIMIT\_ADV\_Y} ({\em gad\_fluxlimit\_adv\_y.F}) |
419 |
|
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$F_y$: {\bf vT} (argument) |
421 |
|
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$V$: {\bf vTrans} (argument) |
423 |
|
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$\tau$: {\bf tracer} (argument) |
425 |
|
426 |
{\em S/R GAD\_FLUXLIMIT\_ADV\_R} ({\em gad\_fluxlimit\_adv\_r.F}) |
427 |
|
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$F_r$: {\bf wT} (argument) |
429 |
|
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$W$: {\bf rTrans} (argument) |
431 |
|
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$\tau$: {\bf tracer} (argument) |
433 |
|
434 |
\end{minipage} } |
435 |
|
436 |
|
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\subsection{Third order direct space time} |
438 |
|
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The direct-space-time method deals with space and time discretization |
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together (other methods that treat space and time separately are known |
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collectively as the ``Method of Lines''). The Lax-Wendroff scheme |
442 |
falls into this category; it adds sufficient diffusion to a second |
443 |
order flux that the forward-in-time method is stable. The upwind |
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biased third order DST scheme is: |
445 |
\begin{eqnarray} |
446 |
F = u \left( \tau_{i-1} |
447 |
+ d_0 (\tau_{i}-\tau_{i-1}) + d_1 (\tau_{i-1}-\tau_{i-2}) \right) |
448 |
& \forall & u > 0 \\ |
449 |
F = u \left( \tau_{i} |
450 |
- d_0 (\tau_{i}-\tau_{i-1}) - d_1 (\tau_{i+1}-\tau_{i}) \right) |
451 |
& \forall & u < 0 |
452 |
\end{eqnarray} |
453 |
where |
454 |
\begin{eqnarray} |
455 |
d_1 & = & \frac{1}{6} ( 2 - |c| ) ( 1 - |c| ) \\ |
456 |
d_2 & = & \frac{1}{6} ( 1 - |c| ) ( 1 + |c| ) |
457 |
\end{eqnarray} |
458 |
The coefficients $d_0$ and $d_1$ approach $1/3$ and $1/6$ respectively |
459 |
as the Courant number, $c$, vanishes. In this limit, the conventional |
460 |
third order upwind method is recovered. For finite Courant number, the |
461 |
deviations from the linear method are analogous to the diffusion added |
462 |
to centered second order advection in the Lax-Wendroff scheme. |
463 |
|
464 |
The DST3 method described above must be used in a forward-in-time |
465 |
manner and is stable for $0 \le |c| \le 1$. Although the scheme |
466 |
appears to be forward-in-time, it is in fact third order in time and |
467 |
the accuracy increases with the Courant number! For low Courant |
468 |
number, DST3 produces very similar results (indistinguishable in |
469 |
Fig.~\ref{fig:advect-1d-lo}) to the linear third order method but for |
470 |
large Courant number, where the linear upwind third order method is |
471 |
unstable, the scheme is extremely accurate |
472 |
(Fig.~\ref{fig:advect-1d-hi}) with only minor overshoots. |
473 |
|
474 |
\fbox{ \begin{minipage}{4.75in} |
475 |
{\em S/R GAD\_DST3\_ADV\_X} ({\em gad\_dst3\_adv\_x.F}) |
476 |
|
477 |
$F_x$: {\bf uT} (argument) |
478 |
|
479 |
$U$: {\bf uTrans} (argument) |
480 |
|
481 |
$\tau$: {\bf tracer} (argument) |
482 |
|
483 |
{\em S/R GAD\_DST3\_ADV\_Y} ({\em gad\_dst3\_adv\_y.F}) |
484 |
|
485 |
$F_y$: {\bf vT} (argument) |
486 |
|
487 |
$V$: {\bf vTrans} (argument) |
488 |
|
489 |
$\tau$: {\bf tracer} (argument) |
490 |
|
491 |
{\em S/R GAD\_DST3\_ADV\_R} ({\em gad\_dst3\_adv\_r.F}) |
492 |
|
493 |
$F_r$: {\bf wT} (argument) |
494 |
|
495 |
$W$: {\bf rTrans} (argument) |
496 |
|
497 |
$\tau$: {\bf tracer} (argument) |
498 |
|
499 |
\end{minipage} } |
500 |
|
501 |
|
502 |
\subsection{Third order direct space time with flux limiting} |
503 |
|
504 |
The overshoots in the DST3 method can be controlled with a flux limiter. |
505 |
The limited flux is written: |
506 |
\begin{equation} |
507 |
F = |
508 |
\frac{1}{2}(u+|u|)\left( \tau_{i-1} + \psi(r^+)(\tau_{i} - \tau_{i-1} )\right) |
509 |
+ |
510 |
\frac{1}{2}(u-|u|)\left( \tau_{i-1} + \psi(r^-)(\tau_{i} - \tau_{i-1} )\right) |
511 |
\end{equation} |
512 |
where |
513 |
\begin{eqnarray} |
514 |
r^+ & = & \frac{\tau_{i-1} - \tau_{i-2}}{\tau_{i} - \tau_{i-1}} \\ |
515 |
r^- & = & \frac{\tau_{i+1} - \tau_{i}}{\tau_{i} - \tau_{i-1}} |
516 |
\end{eqnarray} |
517 |
and the limiter is the Sweby limiter: |
518 |
\begin{equation} |
519 |
\psi(r) = \max[0, \min[\min(1,d_0+d_1r],\frac{1-c}{c}r ]] |
520 |
\end{equation} |
521 |
|
522 |
\fbox{ \begin{minipage}{4.75in} |
523 |
{\em S/R GAD\_DST3FL\_ADV\_X} ({\em gad\_dst3\_adv\_x.F}) |
524 |
|
525 |
$F_x$: {\bf uT} (argument) |
526 |
|
527 |
$U$: {\bf uTrans} (argument) |
528 |
|
529 |
$\tau$: {\bf tracer} (argument) |
530 |
|
531 |
{\em S/R GAD\_DST3FL\_ADV\_Y} ({\em gad\_dst3\_adv\_y.F}) |
532 |
|
533 |
$F_y$: {\bf vT} (argument) |
534 |
|
535 |
$V$: {\bf vTrans} (argument) |
536 |
|
537 |
$\tau$: {\bf tracer} (argument) |
538 |
|
539 |
{\em S/R GAD\_DST3FL\_ADV\_R} ({\em gad\_dst3\_adv\_r.F}) |
540 |
|
541 |
$F_r$: {\bf wT} (argument) |
542 |
|
543 |
$W$: {\bf rTrans} (argument) |
544 |
|
545 |
$\tau$: {\bf tracer} (argument) |
546 |
|
547 |
\end{minipage} } |
548 |
|
549 |
|
550 |
\subsection{Multi-dimensional advection} |
551 |
|
552 |
\begin{figure} |
553 |
\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-lo-diag.eps}} |
554 |
\caption{ |
555 |
Comparison of advection schemes in two dimensions; diagonal advection |
556 |
of a resolved Gaussian feature. Courant number is 0.01 with |
557 |
30$\times$30 points and solutions are shown for T=1/2. White lines |
558 |
indicate zero crossing (ie. the presence of false minima). The left |
559 |
column shows the second order schemes; top) centered second order with |
560 |
Adams-Bashforth, middle) Lax-Wendroff and bottom) Superbee flux |
561 |
limited. The middle column shows the third order schemes; top) upwind |
562 |
biased third order with Adams-Bashforth, middle) third order direct |
563 |
space-time method and bottom) the same with flux limiting. The top |
564 |
right panel shows the centered fourth order scheme with |
565 |
Adams-Bashforth and right middle panel shows a fourth order variant on |
566 |
the DST method. Bottom right panel shows the Superbee flux limiter |
567 |
(second order) applied independently in each direction (method of |
568 |
lines). |
569 |
\label{fig:advect-2d-lo-diag} |
570 |
} |
571 |
\end{figure} |
572 |
|
573 |
\begin{figure} |
574 |
\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-mid-diag.eps}} |
575 |
\caption{ |
576 |
Comparison of advection schemes in two dimensions; diagonal advection |
577 |
of a resolved Gaussian feature. Courant number is 0.27 with |
578 |
30$\times$30 points and solutions are shown for T=1/2. White lines |
579 |
indicate zero crossing (ie. the presence of false minima). The left |
580 |
column shows the second order schemes; top) centered second order with |
581 |
Adams-Bashforth, middle) Lax-Wendroff and bottom) Superbee flux |
582 |
limited. The middle column shows the third order schemes; top) upwind |
583 |
biased third order with Adams-Bashforth, middle) third order direct |
584 |
space-time method and bottom) the same with flux limiting. The top |
585 |
right panel shows the centered fourth order scheme with |
586 |
Adams-Bashforth and right middle panel shows a fourth order variant on |
587 |
the DST method. Bottom right panel shows the Superbee flux limiter |
588 |
(second order) applied independently in each direction (method of |
589 |
lines). |
590 |
\label{fig:advect-2d-mid-diag} |
591 |
} |
592 |
\end{figure} |
593 |
|
594 |
\begin{figure} |
595 |
\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-hi-diag.eps}} |
596 |
\caption{ |
597 |
Comparison of advection schemes in two dimensions; diagonal advection |
598 |
of a resolved Gaussian feature. Courant number is 0.47 with |
599 |
30$\times$30 points and solutions are shown for T=1/2. White lines |
600 |
indicate zero crossings and initial maximum values (ie. the presence |
601 |
of false extrema). The left column shows the second order schemes; |
602 |
top) centered second order with Adams-Bashforth, middle) Lax-Wendroff |
603 |
and bottom) Superbee flux limited. The middle column shows the third |
604 |
order schemes; top) upwind biased third order with Adams-Bashforth, |
605 |
middle) third order direct space-time method and bottom) the same with |
606 |
flux limiting. The top right panel shows the centered fourth order |
607 |
scheme with Adams-Bashforth and right middle panel shows a fourth |
608 |
order variant on the DST method. Bottom right panel shows the Superbee |
609 |
flux limiter (second order) applied independently in each direction |
610 |
(method of lines). |
611 |
\label{fig:advect-2d-hi-diag} |
612 |
} |
613 |
\end{figure} |
614 |
|
615 |
|
616 |
|
617 |
In many of the aforementioned advection schemes the behavior in |
618 |
multiple dimensions is not necessarily as good as the one dimensional |
619 |
behavior. For instance, a shape preserving monotonic scheme in one |
620 |
dimension can have severe shape distortion in two dimensions if the |
621 |
two components of horizontal fluxes are treated independently. There |
622 |
is a large body of literature on the subject dealing with this problem |
623 |
and among the fixes are operator and flux splitting methods, corner |
624 |
flux methods and more. We have adopted a variant on the standard |
625 |
splitting methods that allows the flux calculations to be implemented |
626 |
as if in one dimension: |
627 |
\begin{eqnarray} |
628 |
\tau^{n+1/3} & = & \tau^{n} |
629 |
- \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n}) |
630 |
+ \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\ |
631 |
\tau^{n+2/3} & = & \tau^{n+1/3} |
632 |
- \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) \\ |
634 |
\tau^{n+3/3} & = & \tau^{n+2/3} |
635 |
- \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) |
637 |
\end{eqnarray} |
638 |
|
639 |
In order to incorporate this method into the general model algorithm, |
640 |
we compute the effective tendency rather than update the tracer so |
641 |
that other terms such as diffusion are using the $n$ time-level and |
642 |
not the updated $n+3/3$ quantities: |
643 |
\begin{equation} |
644 |
G^{n+1/2}_{adv} = \frac{1}{\Delta t} ( \tau^{n+3/3} - \tau^{n} ) |
645 |
\end{equation} |
646 |
So that the over all time-stepping looks likes: |
647 |
\begin{equation} |
648 |
\tau^{n+1} = \tau^{n} + \Delta t \left( G^{n+1/2}_{adv} + G_{diff}(\tau^{n}) + G^{n}_{forcing} \right) |
649 |
\end{equation} |
650 |
|
651 |
\fbox{ \begin{minipage}{4.75in} |
652 |
{\em S/R GAD\_ADVECTION} ({\em gad\_advection.F}) |
653 |
|
654 |
$\tau$: {\bf Tracer} (argument) |
655 |
|
656 |
$G^{n+1/2}_{adv}$: {\bf Gtracer} (argument) |
657 |
|
658 |
$F_x, F_y, F_r$: {\bf af} (local) |
659 |
|
660 |
$U$: {\bf uTrans} (local) |
661 |
|
662 |
$V$: {\bf vTrans} (local) |
663 |
|
664 |
$W$: {\bf rTrans} (local) |
665 |
|
666 |
\end{minipage} } |
667 |
|
668 |
\begin{figure} |
669 |
\resizebox{3.5in}{!}{\includegraphics{s_algorithm/figs/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} |
676 |
\label{sec:tracer_advection_schemes} |
677 |
\begin{rawhtml} |
678 |
<!-- CMIREDIR:comparison_of_advection_schemes: --> |
679 |
\end{rawhtml} |
680 |
|
681 |
\begin{table}[htb] |
682 |
\centering |
683 |
{\small |
684 |
\begin{tabular}[htb]{|l|c|c|c|c|l|} |
685 |
\hline |
686 |
Advection Scheme & code & use & use Multi- & Stencil & comments \\ |
687 |
& & A.B. & dimension & (1 dim) & \\ |
688 |
\hline \hline |
689 |
$1^{rst}$order upwind & 1 & No & Yes & 3 pts & linear/$\tau$, non-linear/v\\ |
690 |
\hline |
691 |
centered $2^{nd}$order & 2 & Yes & No & 3 pts & linear \\ |
692 |
\hline |
693 |
$3^{rd}$order upwind & 3 & Yes & No & 5 pts & linear/$\tau$\\ |
694 |
\hline |
695 |
centered $4^{th}$order & 4 & Yes & No & 5 pts & linear \\ |
696 |
\hline \hline |
697 |
$2^{nd}$order DST (Lax-Wendroff) & 20 & |
698 |
No & Yes & 3 pts & linear/$\tau$, non-linear/v\\ |
699 |
\hline |
700 |
$3^{rd}$order DST & 30 & No & Yes & 5 pts & linear/$\tau$, non-linear/v\\ |
701 |
\hline |
702 |
$2^{nd}$order-moment Prather & 80 & No & Yes & ~ & ~ \\ |
703 |
\hline \hline |
704 |
$2^{nd}$order Flux Limiters & 77 & No & Yes & 5 pts & non-linear \\ |
705 |
\hline |
706 |
$3^{nd}$order DST Flux limiter & 33 & No & Yes & 5 pts & non-linear \\ |
707 |
\hline |
708 |
$2^{nd}$order-moment Prather w. limiter & 81 & No & Yes & ~ & ~ \\ |
709 |
\hline |
710 |
piecewise parabolic w. ``null'' limiter & 40 & No & Yes & ~ & ~ \\ |
711 |
\hline |
712 |
piecewise parabolic w. ``mono'' limiter & 41 & No & Yes & ~ & ~ \\ |
713 |
\hline |
714 |
piecewise quartic w. ``null'' limiter & 50 & No & Yes & ~ & ~ \\ |
715 |
\hline |
716 |
piecewise quartic w. ``mono'' limiter & 51 & No & Yes & ~ & ~ \\ |
717 |
\hline |
718 |
piecewise quartic w. ``weno'' limiter & 52 & No & Yes & ~ & ~ \\ |
719 |
\hline |
720 |
$7^{nd}$order one-step method & 7 & No & Yes & ~ & ~ \\ |
721 |
with Monotonicity Preserving Limiter & ~ & ~ & ~ & ~ & ~ \\ |
722 |
\hline |
723 |
|
724 |
\end{tabular} |
725 |
} |
726 |
\caption{Summary of the different advection schemes available in MITgcm. |
727 |
``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time. |
728 |
The code corresponds to the number used to select the corresponding |
729 |
advection scheme in the parameter file (e.g., {\bf tempAdvScheme}=3 in |
730 |
file {\em data} selects the $3^{rd}$ order upwind advection scheme |
731 |
for temperature). |
732 |
} |
733 |
\label{tab:advectionShemes_summary} |
734 |
\end{table} |
735 |
|
736 |
|
737 |
Figs.~\ref{fig:advect-2d-lo-diag}, \ref{fig:advect-2d-mid-diag} and |
738 |
\ref{fig:advect-2d-hi-diag} show solutions to a simple diagonal |
739 |
advection problem using a selection of schemes for low, moderate and |
740 |
high Courant numbers, respectively. The top row shows the linear |
741 |
schemes, integrated with the Adams-Bashforth method. Theses schemes |
742 |
are clearly unstable for the high Courant number and weakly unstable |
743 |
for the moderate Courant number. The presence of false extrema is very |
744 |
apparent for all Courant numbers. The middle row shows solutions |
745 |
obtained with the unlimited but multi-dimensional schemes. These |
746 |
solutions also exhibit false extrema though the pattern now shows |
747 |
symmetry due to the multi-dimensional scheme. Also, the schemes are |
748 |
stable at high Courant number where the linear schemes weren't. The |
749 |
bottom row (left and middle) shows the limited schemes and most |
750 |
obvious is the absence of false extrema. The accuracy and stability of |
751 |
the unlimited non-linear schemes is retained at high Courant number |
752 |
but at low Courant number the tendency is to loose amplitude in sharp |
753 |
peaks due to diffusion. The one dimensional tests shown in |
754 |
Figs.~\ref{fig:advect-1d-lo} and \ref{fig:advect-1d-hi} showed this |
755 |
phenomenon. |
756 |
|
757 |
Finally, the bottom left and right panels use the same advection |
758 |
scheme but the right does not use the multi-dimensional method. At low |
759 |
Courant number this appears to not matter but for moderate Courant |
760 |
number severe distortion of the feature is apparent. Moreover, the |
761 |
stability of the multi-dimensional scheme is determined by the maximum |
762 |
Courant number applied of each dimension while the stability of the |
763 |
method of lines is determined by the sum. Hence, in the high Courant |
764 |
number plot, the scheme is unstable. |
765 |
|
766 |
With many advection schemes implemented in the code two questions |
767 |
arise: ``Which scheme is best?'' and ``Why don't you just offer the |
768 |
best advection scheme?''. Unfortunately, no one advection scheme is |
769 |
``the best'' for all particular applications and for new applications |
770 |
it is often a matter of trial to determine which is most |
771 |
suitable. Here are some guidelines but these are not the rule; |
772 |
\begin{itemize} |
773 |
\item If you have a coarsely resolved model, using a |
774 |
positive or upwind biased scheme will introduce significant diffusion |
775 |
to the solution and using a centered higher order scheme will |
776 |
introduce more noise. In this case, simplest may be best. |
777 |
\item If you have a high resolution model, using a higher order |
778 |
scheme will give a more accurate solution but scale-selective |
779 |
diffusion might need to be employed. The flux limited methods |
780 |
offer similar accuracy in this regime. |
781 |
\item If your solution has shocks or propagating fronts then a |
782 |
flux limited scheme is almost essential. |
783 |
\item If your time-step is limited by advection, the multi-dimensional |
784 |
non-linear schemes have the most stability (up to Courant number 1). |
785 |
\item If you need to know how much diffusion/dissipation has occurred you |
786 |
will have a lot of trouble figuring it out with a non-linear method. |
787 |
\item The presence of false extrema is non-physical and this alone is the |
788 |
strongest argument for using a positive scheme. |
789 |
\end{itemize} |