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In summary, we have defined the following terms from the analysis of
finite difference schemes for the linear shift-invariant case with
constant sampling rates:
- PDE well posed
PDE at least marginally stable
- FDS consistent
FDS shift operator PDE operator as
- FDS stable
stable or marginally stable as a digital filter
- FDS strictly stable
stable as a digital filter
- FDS marginally stable
marginally stable as a digital filter
Finally, the Lax-Richtmyer equivalence theorem establishes that well
posed + consistency + stability implies convergence, where, as defined
in §L.2 above, convergence means that solutions of the
FDS approach corresponding solutions of the PDE as .
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