# Time Domain Room Acoustic Solver with Fourth-Order Explicit FEM Using Modified Time Integration
## 1 Introduction
### Room Acoustic Solver Using Explicit TD-FEM and Contributions of the Present Paper  
The room acoustic solver using explicit TD-FEM [24; 25; 26] is based on simultaneous first-order ordinary differential equations (ODEs) equivalent to a second-order ODE derived by application of the Galerkin method to a wave equation. It achieves fourth-order accuracy in both space and time. It also uses dispersion-reduced FEs as in the implicit TD-FEM described above. The fully explicit and stable scheme is achieved by introducing a matrix lumping into a dissipation term [25]. Earlier study [24] revealed that the explicit TD-FEM has better performance than the fourth-order accurate implicit TD-FEM in terms of computational times to achieve similar accurate results. In subsequent study [26], an extended reaction model for permeable membrane absorbers is implemented to increase its applicability. However, at this stage, the explicit method has a shortcoming by which fourth-order accuracy in time is maintained for the case using only square or cubic FEs. As a result, staircase approximation is introduced into room shape modeling. In general, room acoustic simulations address sound propagation in complex shaped rooms. When using staircase approximation in room shape modeling of complex shaped rooms, sufficiently small size of elements must be used to maintain the approximation error acceptable. That treatment increases computational costs significantly. Therefore, introducing the approximation reduces its attractiveness as a room acoustic solver. In addition, use of higher-order accurate time integration methods are essential to perform acoustic simulations at kilohertz frequencies efficiently. The purpose of the present study is to overcome shortcomings inherent in designing a special time integration method suitable for explicit TD-FEM. We propose a novel room acoustic solver using fourth-order accurate explicit TD-FEM in both space and time. Then we demonstrate its practicality as a room acoustic design tool. The proposed explicit TD-FEM can fit room boundaries using irregularly shaped FEs. Moreover, it can eliminate use of the staircase approximation in room shape modeling while maintaining fourth-order accuracy in time without additional computational complexity compared to that achieved using the conventional method. The discretization error characteristics of the proposed method are investigated theoretically and numerically to elucidate its basic performance. Additionally, the effectiveness of the proposed method for practical applications is examined using numerical examples. Notably efficient room acoustic solvers have been produced using the discontinuous Galerkin method [27] and the spectral element method [28]. To reduce dispersion error, they use higher-order elements and Runge-Kutta method, exhibiting its attractive capabilities for room acoustic modeling. As opposed to these higher-order methods, the proposed explicit TD-FEM in the present paper is an attempt to achieve higher order accuracy using low-order linear FEs. An important advantage is that the resulting sparse matrix has narrower bandwidth than those of higher-order methods. Moreover, our designed time-integration method requires less computational complexity than higher-order Runge-Kutta method. Although the present method has a slight dissipation error, the error magnitude is controllable by the time interval used. The present paper is organized as follows. Section 2 introduces the theory of conventional explicit TD-FEM applied to room acoustic simulations, including detailed explanations of its shortcomings which must be overcome. Section 3 presents the theory of the novel explicit TD-FEM. To reduce both spatial and temporal discretization errors, frequency domain and time domain dispersion analyses are performed respectively. Also, a new time marching scheme is derived with a specially designed time integration method. The section also includes derivation of its stability condition.