Finite Difference Schemes

This appendix gives some simplified definitions and results from the subject of finite difference schemes for numerically solving partial differential equations. An excellent reference on this subject is Strikwerda [495], and a recent dissertation in this area is that of Bilbao [54,55].

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 [460]. For a more general and traditional treatment of PDEs and their associated finite difference schemes, see, e.g., [495].

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Subsections

• Finite Difference Schemes
• Convergence
  • Consistency
  • Well Posed Initial-Value Problem
    • A Class of Well Posed Damped PDEs
    • Proof that the Third-Order Time Derivative is Ill Posed
  • 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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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.