On the numerical solution of the 2d wave equation with compact fdtd schemes

This paper discusses compact-stencil nite difference time domain (FDTD) schemes for approximating the 2D wave equation in the context of digital audio. Stability, accuracy, and ef ciency are investigated and new ways of viewing and interpreting the results are discussed. It is shown that if a tight accuracy constraint is applied, implicit schemes outperform explicit schemes. The paper also discusses the relevance to digital waveguide mesh modelling, and highlights the optimally ef cient explicit scheme.

[1]  Allen Taflove,et al.  Computational Electrodynamics the Finite-Difference Time-Domain Method , 1995 .

[2]  Stefan Bilbao.  Parameterized families of finite difference schemes for the wave equation , 2004 .

[3]  R. J. Kruhlak,et al.  Dispersion of homogeneous and inhomogeneous waves in the Yee finite-difference time-domain grid , 2001 .

[4]  R. F. Warming,et al.  The modified equation approach to the stability and accuracy analysis of finite-difference methods , 1974 .

[5]  Pierre Ruiz,et al.  Synthesizing Musical Sounds by Solving the Wave Equation for Vibrating Objects: Part 2 , 1971 .

[6]  Davide Rocchesso,et al.  Signal-theoretic characterization of waveguide mesh geometries for models of two-dimensional wave propagation in elastic media , 2001, IEEE Trans. Speech Audio Process..

[7]  Vesa Välimäki,et al.  Interpolated and warped 2-D digital waveguide mesh algorithms , 2000 .

[8]  Julius O. Smith,et al.  Finite difference schemes and digital waveguide networks for the wave equation: stability, passivity, and numerical dispersion , 2003, IEEE Trans. Speech Audio Process..

[9]  Alfred Fettweis,et al.  Multidimensional wave digital filters for discrete‐time modelling of Maxwell's equations , 1992 .

[10]  S.,et al.  Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media , 1966 .

[11]  Vesa Välimäki,et al.  Reducing the dispersion error in the digital waveguide mesh using interpolation and frequency-warping techniques , 2000, IEEE Trans. Speech Audio Process..

[12]  S. Van Duyne,et al.  The 2-D digital waveguide mesh , 1993, Proceedings of IEEE Workshop on Applications of Signal Processing to Audio and Acoustics.

[13]  Maarten van Walstijn,et al.  On-Line Simulation of 2D Resonators with Reduced Dispersion Error using Compact Implicit Finite Difference Methods , 2007, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07.

[14]  S. Bilbao.  Wave and Scattering Methods for Numerical Simulation , 2004 .

[15]  D. Botteldooren.  Finite‐difference time‐domain simulation of low‐frequency room acoustic problems , 1995 .

[16]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .