# Time Domain Room Acoustic Solver with Fourth-Order Explicit FEM Using Modified Time Integration
## 1 Introduction  
### Study Background  
For room acoustic design, it is crucially important to predict accurate impulse responses to provide comfortable acoustic environments necessary for various rooms such as concert halls, offices, and classrooms. Using impulse responses, one can design room acoustics via visualization and auralization of sound fields as well as calculating room acoustic parameters such as reverberation times and the speech transmission index. Computer simulation methods are indispensable tools for room acoustic modeling because they can virtually simulate acoustics in architectural spaces. Furthermore, simulation can facilitate parametric studies more readily than scale model experiments for designing basic room shapes and interior finishes that is, selection of acoustical absorptive or reflective materials [1]. Geometrical acoustics simulation methods have flourished since their originalproposition in the 1960s [2; 3] to today, as a practical room acoustic design tool [4; 5]. Geometrical acoustics methods can be undertaken with low computational costs through simplified approximation of wave phenomena by which wave propagation is modeled as a ray propagation. Because of such approximation, geometrical acoustics methods have less ability to accommodate or reflect wave phenomena at low and mid-range frequencies, although studies for alleviating these difficulties have been conducted, for example, the implementations of diffraction model [6] and scattering coefficients [7]. Nevertheless, recent progress in computer technology has increased the applicability of inherently accurate wave acoustic methods; those circumstances promote its use in practical applications [1]. Wave acoustics methods are classified into frequency domain methods and time domain methods. The latter, time domain methods, can calculate room impulse responses directly. They have attracted more intense interest for application as room acoustic solvers. In general time domain wave acoustics methods, the wave equation or its first-order forms, that is, the continuity equation and Euler equation, are discretized in both space and time. The discretized equation is solved numerically with appropriate initial and boundary conditions. They are inherently accurate, including all wave phenomena naturally. However, wave acoustics methods demand considerable computational effort, entailing enormous memory requirements and long computational timed, especially for acoustic simulations of actual rooms at kilohertz frequencies because one must control the inherent discretization error, so-called dispersion error, to obtain reliable results. The dispersion error is frequency and directional angle dependent error with respect to sound speed caused by both spatial and temporal discretization. Fundamentally, a sufficiently finer spatial discretization than the wavelength of frequency to be analyzed and the controlling the phase error coming from time discretization must be achieved to reduce the numerical dispersion. However, that requirement engenders large-scale problems that entail many degrees of freedom (DOF) and time steps. Consequently, various efficient time domain methods have been developed to date for practical application of wave acoustic methods. The finite difference time domain (FDTD) method [8; 9; 10; 11; 12] is the most popular and well-used method. The FDTD method models sound propagation in rooms by discretizing the partial differential equation straightforwardly using finite difference approximation in both space and time. The implementation is simple and attractive for computational efficiency, but it often suffers from both dispersion error and approximation error in room shapes coming from staircase approximation. To alleviate these shortcomings, higher-order accurate methods have been proposed based on compact difference method [11], modified equation method [12], and application of unstructured grid based on finite volume modeling [13].