Summary of Well Posedness, Consistency, Stability, and Convergence

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 §N.2 above, convergence means that solutions of the FDS approach corresponding solutions of the PDE as .

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.