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 Science Queen’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 ｿnite difference (FD) technique is proposed to reduce the dispersion error remarkably. In par- ticular, a computationally efｿcient method that allows solving 2D implicit problems with a set of three-diagonal equations, namely the alternating direction implicit is discussed. Efｿ- cient equation factorisation together with optimally matched free parameters allows more accurate simulation for wider frequency ranges. With the use of this technique, the disper- sion error is limited to 1.1% within the bandwidth up to half of the Nyquist frequency. The compact implicit scheme is com- pared to compact explicit FD schemes in terms of numerical dispersion error, membrane impulse response, and computa- tional cost. Index Terms— Acoustic propagation, acoustic signal pro- cessing, ｿnite difference methods, FDTD methods, waveg- uides 1. INTRODUCTION Simulation of vibrating systems such as membranes and plates requires numerically solving the two-dimensional wave equa- tion, for which ｿnite difference methods can be used [1]. These methods are also suited for three-dimensional room auralisation purposes [2]. In reality, the sound wave propa- gation speed in such resonators is independent of propagation direction. An unwanted side effect of using numerical ｿnite difference (FD) schemes is that they introduce a direction- dependent dispersion error. In the original rectangular two- dimensional digital waveguide mesh (DWG), which is math- ematically equivalent to a compact explicit FD scheme, this error is particularly severe in axial directions [3]. In order to make the error of the rectangular mesh homogeneous in all directions, interpolation techniques have been introduced [4], which make the dispersion error nearly direction-independent This research has been supported by the European Social Fund. [5]. However, a signiｿcant frequency-dependent dispersion error remains. In order to compensate for the directionally in- dependent dispersion error, Savioja and V¨ alim¨ aki introduced a frequency warping technique [6], with which the remaining maximum dispersion error of the interpolated digital waveg- uide mesh (IDWM) is reduced to 1.2% [5]. However, the main disadvantage of the frequency warping technique is that pre- and post-processing has to be executed off-line, which makes this method unsuitable for on-line applications. An on- line frequency warping technique has been treated in [7] for a triangular mesh, but this approach brings about the necessity of increasing the sampling frequency and low-pass ｿltering of the output signal. Another approach relies on adding a reac- tive load to each grid point [8]. The aim of this paper is to propose the use of an FD method with signiｿcantly reduced frequency error without much increase in computational cost, which would be suitable for on-line simulations. Consequently, such an FD method would be applicable to interactive applications. 2. COMPACT IMPLICIT FINITE DIFFERENCE SCHEMES Many higher-order accuracy FD schemes are difｿcult to de- sign mainly due to ensuring stability near boundaries and low operation count. In particular, ‘higher-order large star sys- tems’ rely on looking at neighbouring nodes more distant from an update point; therefore these systems are inconvenient due to complicated update formulae near boundaries [9]. In order to reduce the numerical error, the use of a com- pact implicit method is proposed, as it achieves the highest order of accuracy on the smallest mesh system [9]. The com- pact implicit approximation to the wave equation relies on updating the node data based on both the neighbouring points in time and space domain, as shown in Figure 1(c). This approach is similar to space interpolation in the interpolated digital waveguide mesh, but in this case all the nearest neigh- bours in time domain are also included in the update formula. Consequently, a higher accuracy can be achieved for a wider I  285 1424407281/07/$20.00 ©2007 IEEE ICASSP 2007