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The MP5 scheme of Suresh and Huynh \cite{Suresh:1997} is based on the keen observation that three points cannot distinguish between extrema and discontinuities. The functional form of the limiter is not as simple as the three-point schemes described above, so we refer the reader to the original paper or the FDS source code for details. But the basic idea behind the method is to use a five-point stencil, three upwind and two downwind, to reconstruct the cell face value, considering both accuracy and monotonicity-preserving constraints. An additional benefit of the MP5 scheme is that it was designed specifically with strong stability-preserving (SSP) Runge-Kutta time discretizations in mind. The predictor-corrector scheme used by FDS is similar to the second-order SSP scheme described in \cite{Gottlieb:2001}.
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\subsubsection{Notes on Implementation}
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In practice, we set $r=0$ initially and only compute $r$ if the denominator is not zero. Note that for $\delta\phi_{loc}=0$, it does not matter which limiter is used: all the limiters yield the same scalar face value. For CHARM, we set both $r=0$ and $B=0$ initially and only compute $B$ if $r>0$ (this requires data variations to have the same sign). Otherwise, CHARM reduces to Godunov's scheme.
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The Central Difference, Godunov, and MINMOD limiters are included for completeness, debugging, and educational purposes. These schemes have little utility for typical FDS applications.
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The Central Differenceand Godunov limiters are included for completeness, debugging, and educational purposes. These schemes have little utility for typical FDS applications.
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\subsubsection{Dealing with Variable Molecular Weights}
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%\subsubsection{Dealing with Variable Molecular Weights}
%% leave this here for a moment as a reminder to write up the constant limiter coefficient method we are now using.
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The above condition is automatically satisfied in the cases of using Godunov or Central differencing or in the case of binary flow (two species). However, if we apply a second-order flux limiter, such as Superbee or CHARM, independently to each species in a multi-component (three or more species) flow with variable molecular weights, then this condition is easily violated.
% The above condition is automatically satisfied in the cases of using Godunov or Central differencing or in the case of binary flow (two species). However, if we apply a second-order flux limiter, such as Superbee or CHARM, independently to each species in a multi-component (three or more species) flow with variable molecular weights, then this condition is easily violated.
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To handle this situation, in LES mode, FDS will apply a correction to the most abundant species locally. We first compute the flux-limited face values of the mass density over the mixture-average molecular weight. Then we compute flux-limited face values of the species densities. Finally, the error in Eq.~(\ref{eq:rho_mw}) is absorbed into the most abundant species locally,
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\section{Flux Limiters}
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\label{info:flux_limiters}
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FDS employs \emph{total variation diminishing} (TVD) schemes for scalar transport. The default for VLES (FDS default \ct{SIMULATION_MODE}) is Superbee \cite{Roe:1986}, so chosen because this scheme does the best job preserving the scalar variance in highly turbulent flows with coarse grid resolution. The default scheme for DNS and LES is CHARM \cite{Zhou:1995} because the gradient steepening used in Superbee forces a stair step pattern at high resolution, while CHARM is convergent. A few other schemes (including Godunov and central differencing) are included for completeness; more details can be found in the Tech Guide \cite{FDS_Tech_Guide}. Table \ref{tab:flux_limiters} below shows the character strings which may be used to invoke the various limiter schemes.
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FDS employs \emph{total variation diminishing} (TVD) schemes for scalar transport. The default for VLES (FDS default \ct{SIMULATION_MODE}) is Superbee \cite{Roe:1986}, so chosen because this scheme does the best job preserving the scalar variance in highly turbulent flows with coarse grid resolution. The default scheme for DNS and LES is CHARM \cite{Zhou:1995} because the gradient steepening used in Superbee forces a stair step pattern at high resolution, while CHARM is convergent. Godunov and central differencing are included for completeness; more details can be found in the Tech Guide \cite{FDS_Tech_Guide}. Table \ref{tab:flux_limiters} below shows the character strings which may be used to invoke the various limiter schemes.
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\label{fig_soborot_square_wave}
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\end{figure}
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For the cosine wave initial condition the derivatives of the scalar field are continuous. Therefore, we see ${\cal O}(\delta x^2)$ convergence of the CHARM and MP5 schemes, as shown in Fig.~\ref{fig_soborot_cos_wave}. Superbee shows smaller error at coarse resolution, but the gradient steepening degenerates its accuracy at higher resolutions---hence CHARM is selected for LES and DNS.
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For the cosine wave initial condition the derivatives of the scalar field are continuous. Therefore, we see ${\cal O}(\delta x^2)$ convergence of the CHARM scheme, as shown in Fig.~\ref{fig_soborot_cos_wave}. Superbee shows smaller error at coarse resolution, but the gradient steepening degenerates its accuracy at higher resolutions---hence CHARM is selected for LES and DNS.
\caption[Solid body rotation cosine wave convergence]{Solid body rotation cosine wave solution convergence. Scalar fields along upper-left diagonal for Superbee (upper-left), CHARM (upper-right), and MP5 (lower-left). (Lower-right) L2 Error.}
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\caption[Solid body rotation cosine wave convergence]{Solid body rotation cosine wave solution convergence. Scalar fields along upper-left diagonal for Superbee (upper-left) and CHARM (upper-right). (Lower-left) L2 Error.}
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\label{fig_soborot_cos_wave}
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\end{figure}
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\subsection{Temperature Lower Bound in a Fire Simulation (\texorpdfstring{\ct{tmp_lower_limit}}{tmp\_lower\_limit})}
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\label{tmp_lower_limit}
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A 2-D methane-air diffusion flame calculation is run three different ways to check that the temperature does not fall below the ambient (20~$^\circ$C). The results are shown in Fig.~\ref{tmp_lower_limit_fig}. The upper left plot shows the global minimum temperature for the case where the reaction stoichiometry is specified explicitly. The upper right plot is for the case where the initial time step is lowered to 0.001~s. The lower plot is for the case where the simple chemistry model is used.
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A 2-D methane-air diffusion flame calculation is run three different ways to check that the temperature does not fall below the ambient (20~$^\circ$C). The results are shown in Fig.~\ref{tmp_lower_limit_fig}. The left plot shows the global minimum temperature for the case where the reaction stoichiometry is specified explicitly. Theright plot is for the case where the simple chemistry model is used.
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