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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.
\caption[JIA A 1310 Facade, position of heat flux gauges]{Positions of the heat flux gauges on the exterior wall of the JIS~A~1310 experiments. The wall is 1.82~m wide and 4.095~m tall. The window is 0.91~m wide by 0.91 tall and 0.455~m above the floor.}
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\caption[JIS Facade, position of heat flux gauges]{Positions of the heat flux gauges on the exterior wall of the JIS~A~1310 experiments. The wall is 1.82~m wide and 4.095~m tall. The window is 0.91~m wide by 0.91 tall and 0.455~m above the floor.}
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