Spectral and Pseudospectral Properties of Finite Difference Models Used in Audio and Room Acoustics

Abstract

Finite difference solutions to the wave equation are simple and flexible modeling tools for approximating physical systems in audio and room acoustics. Each model is characterized by a matrix operator and the time-stepping solution by a sequence of powers of the matrix. Spectral decomposition of representative matrices provide some practical insight into solution behavior and in some cases stability. In addition to computed eigenvalue spectra, pseudospectra provide a description of numerical amplification due to rounding errors in floating point arithmetic. The matrix analysis also shows that certain boundary implementations in non-cuboid geometries can be unstable despite satisfying conditions derived from von Neumann and normal mode analyses.