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 [53,55] and Strikwerda [484].

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 [452]. For a more general and traditional treatment of PDEs and their associated finite-difference schemes, see, e.g., [484].

• 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