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This appendix gives some simplified definitions and results from the
subject of *finite-difference schemes* for numerically solving
partial differential equations. Excellent references on this subject
include Bilbao [53,55] and Strikwerda
[481].

The simplifications adopted here are that we will exclude nonlinear
and time-varying partial differential equations (PDEs). We will
furthermore assume constant step-sizes (sampling intervals) when
converting PDEs to finite-difference schemes (FDSs), *i.e.*, sampling
rates along time and space will be constant. Accordingly, we will
assume that all initial conditions are *bandlimited* to less than
half the *spatial* sampling rate, and that all excitations over
time (such as summing input signals or ``moving boundary conditions'')
will be bandlimited to less than half the *temporal* sampling
rate. In short, the simplifications adopted here make the subject of
partial differential equations isomorphic to that of linear systems
theory [449]. For a more general and traditional treatment of
PDEs and their associated finite-difference schemes, see,
*e.g.*, [481].

- Finite-Difference Schemes
- Convergence
- Consistency
- Well Posed Initial-Value Problem
- Stability of a Finite-Difference Scheme
- Lax-Richtmyer equivalence theorem
- Passivity of a Finite-Difference Scheme
- Summary
- Convergence in Audio Applications

- Characteristic Polynomial Equation
- Von Neumann Analysis

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (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.

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