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Summary

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 $ \Leftrightarrow$ PDE at least marginally stable
  • FDS consistent $ \Leftrightarrow$ FDS shift operator $ \to$ PDE operator as $ T,X\to0$
  • FDS stable $ \Leftrightarrow$ stable or marginally stable as a digital filter
  • FDS strictly stable $ \Leftrightarrow$ stable as a digital filter
  • FDS marginally stable $ \Leftrightarrow$ marginally stable as a digital filter
Finally, the Lax-Richtmyer equivalence theorem establishes that well posed + consistency + stability implies convergence, where, as defined in §D.2 above, convergence means that solutions of the FDS approach corresponding solutions of the PDE as $ T,X\to0$.

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Passivity of a Finite-Difference Scheme