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2.18.1 Second order flux limiters

The second order flux limiter method can be cast in several ways but is generally expressed in terms of other flux approximations. For example, in terms of a first order upwind flux and second order Lax-Wendroff flux, the limited flux is given as:

$\displaystyle F = F_1 + \psi(r) F_{LW}$ (2.187)

where $ \psi(r)$ is the limiter function,

$\displaystyle F_1 = u \overline{\tau}^i - \frac{1}{2} \vert u\vert \delta_i \tau$ (2.188)

is the upwind flux,

$\displaystyle F_{LW} = F_1 + \frac{\vert u\vert}{2} (1-c) \delta_i \tau$ (2.189)

is the Lax-Wendroff flux and $ c = \frac{u \Delta t}{\Delta x}$ is the Courant (CFL) number.

The limiter function, $ \psi(r)$ , takes the slope ratio

$\displaystyle r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \tau_{i} - \tau_{i-1} }$ $\displaystyle \forall$ $\displaystyle u > 0$ (2.190)
$\displaystyle r = \frac{ \tau_{i+1} - \tau_{i} }{ \tau_{i} - \tau_{i-1} }$ $\displaystyle \forall$ $\displaystyle u < 0$ (2.191)

as it's argument. There are many choices of limiter function but we only provide the Superbee limiter Roe [1985]:

$\displaystyle \psi(r) = \max[0,\min[1,2r],\min[2,r]]$ (2.192)

\fbox{ \begin{minipage}{4.75in} {\em S/R GAD\_FLUXLIMIT\_ADV\_X} ({\em gad\_flux... ...bf rTrans} (argument) \par $\tau$: {\bf tracer} (argument) \par \end{minipage} }

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Next: 2.18.2 Third order direct Up: 2.18 Non-linear advection schemes Previous: 2.18 Non-linear advection schemes   Contents
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