On the Accuracy of First-Order Numerical Derivatives in Multidimensional Digital Waveguide Mesh Topologies

Time-Domain Simulation of Directive Sources in 3-D Digital Waveguide Mesh-Based Acoustical Models

“…Among these, the uniform rectilinear mesh topology provides a more attractive option in terms of the intuitive positioning of its junctions and the accuracy of numerical derivatives when the mesh density (i.e., the number of junctions per unit volume) is constant [39]. It also allows a more structured memory organization and has a well-documented prospect for software optimization and parallelization [40], [41].…”

“…However, such a smooth bivariate function does not exist due to discontinuities occurring at , and . Therefore an approximate function is selected such that (39) The closed-form definition of the directivity function allows the interpolation of intermediate values between horizontal and vertical planes. In addition, it allows easy rotation of the modeled directivity function in azimuth and elevation.…”

Hexagonal vs. rectilinear grids for explicit finite difference schemes for the two-dimensional wave equation

“…The hexagonal grid has been used in other finite differences approximations [14,15,16,17,18], but has not received much attention in FDTD literature aside from some studies [19,3,7,20,21]. It has received some attention in the Transmission Line Matrix (TLM) [22] and the Digital Waveguide Mesh (DWM) [23,5,24] literatures, both of which are scattering-based methods equivalent to the FDTD method at the passivity limit [7]. In the case of the 7-point hexagonal scheme, however, the more general FDTD method allows for a larger Courant number than the DWM/TLM formulation, resulting in less numerical dispersion [7].…”