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Next: 2.17.3 Third order direct Up: 2.17 Non-linear advection schemes Previous: 2.17.1 Second order flux   Contents

2.17.2 Third order direct space time

The direct-space-time method deals with space and time discretization 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 biased third order DST scheme is:

$\displaystyle F = u \left( \tau_{i-1} + d_0 (\tau_{i}-\tau_{i-1}) + d_1 (\tau_{i-1}-\tau_{i-2}) \right)$ $\displaystyle \forall$ $\displaystyle u > 0$ (2.193)
$\displaystyle F = u \left( \tau_{i} - d_0 (\tau_{i}-\tau_{i-1}) - d_1 (\tau_{i+1}-\tau_{i}) \right)$ $\displaystyle \forall$ $\displaystyle u < 0$ (2.194)

where
$\displaystyle d_1$ $\displaystyle =$ $\displaystyle \frac{1}{6} ( 2 - \vert c\vert ) ( 1 - \vert c\vert )$ (2.195)
$\displaystyle d_2$ $\displaystyle =$ $\displaystyle \frac{1}{6} ( 1 - \vert c\vert ) ( 1 + \vert c\vert )$ (2.196)

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 to centered second order advection in the Lax-Wendroff scheme.

The DST3 method described above must be used in a forward-in-time manner and is stable for $ 0 \le \vert c\vert \le 1$. Although the scheme 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. 2.12) to the linear third order method but for large Courant number, where the linear upwind third order method is unstable, the scheme is extremely accurate (Fig. 2.13) with only minor overshoots.

\fbox{ \begin{minipage}{4.75in} {\em S/R GAD\_DST3\_ADV\_X} ({\em gad\_dst3\_adv... ...bf rTrans} (argument) \par $\tau$: {\bf tracer} (argument) \par \end{minipage} }

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Next: 2.17.3 Third order direct Up: 2.17 Non-linear advection schemes Previous: 2.17.1 Second order flux   Contents
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