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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
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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:
where
The coefficients and approach and respectively
as the Courant number, , 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
. 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.
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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