On-Line Simulation of 2D Resonators with Reduced Dispersion Error using Compact Implicit Finite Difference Methods

Abstract

On-line simulation of 2D resonators with reduced dispersion error using compact implicit finite difference methods Kowalczyk, K., & Van Walstijn, M. (2007). On-line simulation of 2D resonators with reduced dispersion error using compact implicit finite difference methods. [285][286][287][288] ON-LINE SIMULATION OF 2D RESONATORS WITH REDUCED DISPERSION ERROR USING COMPACT IMPLICIT FINITE DIFFERENCE METHODS Konrad Kowalczyk and Maarten van Walstijn Sonic Arts Research Centre School of Electronics, Electrical Engineering and Computer ScienceQueen's University of Belfast, Belfast, Northern Ireland kkowalczyk01@qub.ac.uk, m.vanwalstijn@qub.ac.uk ABSTRACT This paper presents a method for on-line simulation of 2D resonators with reduced direction-dependent frequency error. The use of a compact implicit finite difference (FD) technique is proposed to reduce the dispersion error remarkably. In particular, a computationally efficient method that allows solving 2D implicit problems with a set of three-diagonal equations, namely the alternating direction implicit is discussed. Efficient equation factorisation together with optimally matched free parameters allows more accurate simulation for wider frequency ranges. With the use of this technique, the dispersion error is limited to 1.1% within the bandwidth up to half of the Nyquist frequency. The compact implicit scheme is compared to compact explicit FD schemes in terms of numerical dispersion error, membrane impulse response, and computational cost.

Virtual room acoustics using finite difference methods. How to model and analyse frequency-dependent boundaries?

“…A fourth-order accurate compact implicit scheme [8] with 1800x1400 junctions was used for the implementation of the interior of the room in order to obtain the wavefront that was as flat as possible. The size of the room and the simulation time (2000 samples at the sample rate of 4kHz) were set in such a way that only the reflections from the investigated boundary could reach a receiver position.…”

Formulation of Locally Reacting Surfaces in FDTD/K-DWM Modelling of Acoustic Spaces

“…For 2D/3D rooms, more accurate FD schemes based on the rectilinear mesh topology may be applied to reduce the dispersion error; one good example is the compact implicit FD technique which achieves the highest order of accuracy on the smallest mesh system [4].…”

“…Provided that an unstaggered grid is used, these can in principle be used to compute boundary nodes in all types of FDTD and DWM simulations, including cases where the room interior nodes are computed with an implicit scheme. When alternating direction implicit (ADI) methods are applied, such as in [4], the use of explicit boundary formulations is not merely a choice but a necessity, as the ADI method requires at each time step that the boundary node values are known before solving for the interior nodes.…”

“…In this procedure, simulations are executed using a grid consisting of 1800×1400 nodes. For most simulations, a fourth-order accurate compact implicit ADI scheme [4] was used, but in specific cases the standard leap-frog scheme was applied; the ADI scheme generally gives better reflectance results due to lower numerical dispersion, which helps in creating more plane wavefronts. Furthermore, the ADI scheme was applied for comparison of the 2D and 1D boundary models as it requires shorter simulation time needed for the whole wavelet to arrive at a receiver position than the standard leapfrog scheme room interior implementation.…”

“…Over the past decade, many studies in the area of FDTD and DWM modelling of 2D and 3D acoustic systems have focused on reducing and/or compensating for the dispersion error (see, for example [3,4]). Recently, more attention has been given to the problem of formulating better numerical approximations of boundaries, and in particular the realization of frequency-dependent boundaries.…”