Finite-Difference Schemes

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 [#!BilbaoNSS!#,#!BilbaoTP!#] and Strikwerda
[#!Strikwerda!#].

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 [#!JOSFP!#]. For a more general and traditional treatment of
PDEs and their associated finite-difference schemes, see,
*e.g.*, [#!Strikwerda!#].

- 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

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