Research Article  Open Access
An Unconditionally Stable Method for Solving the Acoustic Wave Equation
Abstract
An unconditionally stable method for solving the timedomain acoustic wave equation using Associated Hermit orthogonal functions is proposed. The secondorder time derivatives in acoustic wave equation are expanded by these orthogonal basis functions. By applying Galerkin temporal testing procedure, the time variable can be eliminated from the calculations. The restriction of CourantFriedrichsLevy (CFL) condition in selecting time step for analyzing thin layer can be avoided. Numerical results show the accuracy and the efficiency of the proposed method.
1. Introduction
Numerical simulation of the acoustic wave equation has been widely used in many areas, such as geophysics, exploration, and ultrasonic detection [1–5]. The finite difference (FD) method in time domain is one of the most widely used methods in solving problems related to acoustics due to its efficiency and simplicity [6–8]. However, as the FD method is an explicit timemarching algorithm, the time step must be limited by the CourantFriedrichsLevy (CFL) condition, which means that the computation time will increase dramatically as the spatial grid becomes small. Hence, some unconditionally stable methods were proposed to solve this problem [9–12]. The main categories of these methods include the alternatingdirection implicit (ADI) method [10] and the orthogonal decomposition method using weighted Laguerre polynomials (WLP) [11]. Recently, Huang et al. [12, 13] developed a new orthogonal decomposition method using Associated Hermit orthogonal functions and have applied it to firstorder Maxwell’s equation.
In this paper, we extended the unconditionally stable method using AH functions to solve the secondorder acoustic wave equation. Firstly, the secondorder time derivatives in the acoustic wave equation are expanded by a set of orthogonal basis functions. Secondly, the time variable can be eliminated from the calculations by using Galerkin’s method. Finally, a set of implicit equations in the whole computational domain is established and solved in the AH domain.
2. Numerical Formulation for the Acoustic Wave Equation
2.1. 1D Acoustic Wave Equation
For simplicity, we consider the 1D acoustic wave equation with an external source in isotropic mediawhere is the sound velocity, is the wave displacement, and is an external source. In order to truncate the computational domain, we choose Mur’s firstorder absorbing boundary condition (ABC) for infinite problem from [14]. At the boundary points or , we have
Consider a set of modified AH orthogonal basis functions given by [12]where is the transformed time variable, is the timetranslating parameter, and is the timescaling parameter. By selecting proper and , the modified AH orthogonal basis functions can be used to approximate the causal displacement bywhere is the th expansion coefficient; is selected according to the bandwidth of the analyzed signal [15].
With the time derivation property of AH functions [15]
We can express the firstorder time derivation and the secondorder time derivation of the displacement aswhere , , and represents the order in the AH domain. The deduction can be found in the Appendix.
Substituting (4) and (7) into (1), we obtain
Based on the orthogonal property of the AH functions, we can obtain equations of the AH expansion coefficients using the temporal Galerkin testing procedure [11]where , . We can partition the spatial domain into . Then the spatial discrete form of (9) at point can be written aswhere is the spatial grid size.
In (10), a set of implicit equations is included in the AH domain. Based on the recurrence relation from the th order coefficients and the th order coefficients to the th order coefficients, we can rewrite (10) in a matrix form:where in a unit matrix. Consider
Here, is a tridiagonal coefficient matrix. and are the tuple representations in AH subspace for the displacement and the external source, respectively. In (11), we can find each point has a relationship with its adjacent two points, and a set of implicit equations can be obtained in the whole computation domain from point to :whereis a tridiagonal matrix with submatrix elements. is a combination of all in the whole computation domain. is a vector related with external source expansion coefficients for all points.
Substitute (4) and (6) into (2) and eliminate the temporal terms to obtain (at )
Using the average techniques, we have
Substituting (16) into (15), we have
Applying (17) into all orders in the AH domain, we can obtainwhere
Using the technique above, the absorbing boundary condition at point can be expressed as
Introducing the boundary condition (18) and (20) into (11), (13) can be replaced bywhere is an adaptive coefficient matrix including absorbing boundary condition and is a coefficient matrix corresponding to .
We can use the lowerupper (LU) decomposition method to decompose and use the backsubstitution method to calculate the coefficients of the displacement for all points. The displacement can be reconstructed finally by using (4).
2.2. 2D Acoustic Wave Equation
Considering the need for practical application, we extend the method to the 2D acoustic wave equation, which is given bywhere is divided into grid.
Substituting (4) and (7) into (22) and rearranging it, we can obtain a matrix form equation which is similar to (11):where is the spatial grid size in direction and is the spatial grid size in direction. In (23), we can find each point has a relationship with its adjacent four points, so matrix is a fivediagonal coefficient matrix in 2D case.
For the absorbing boundary condition, the boundary condition can be obtained by using (15) to (20). But the cornerpoint needs special treatment and the spatial grid size is adjusted to at cornerpoint. For the four cornerpoints, Mur’s absorbing boundary condition can be expressed as
Introducing Mur’s absorbing boundary condition, the coefficient matrix can be adjusted to . Using the lowerupper (LU) decomposition method and the backsubstitution method, the coefficients of all points can be calculated. And the displacement can be reconstructed finally using the method described in 1D case.
3. Numerical Results
3.1. 1D Case
In order to validate the proposed method, a 1D structure constituted by two isotropic media separated by a thin layer ( m, km/s) is analyzed. As shown in Figure 1, the thin layer is located at 60 m, and the total simulation length is 104 m. Displacement at two points (38 m) and (78.1 m) is observed. The sound velocity in media A is 2.5 km/s, and the sound velocity in media B is 3.7 km/s. Nonuniform spatial grids are used. In the thin layer m; in other parts m.
The external source is located at m and its expression is , where Hz. Time support for the simulation is 0.16 s. To expand in AH domain, we select , , and . Due to the limitation of the CFL condition, the time step is set as s for the FD method under the condition of m. Time step is not involved in AH domain computation. However, for AH decomposition, we still need to specify a sample interval; the time step s is good enough. The absolute error is defined as for comparison of the two methods. is the reference data obtained from the FD method with m, and is the numerical result obtained from the FD method ( m) and the proposed method ( m). The numerical results at and their errors are shown in Figure 2(a). Figure 2(b) is the numerical results at . From Figure 2, it can be seen that the proposed method has a good agreement with the FD method. The absolute error of the proposed method ( m) is much lower than that of the FD method ( m) at both and .
(a)
(b)
Table 1 shows the comparison of CPU time between the proposed method and the FD method. Under the condition of m, the time step of the proposed method is 1000 times as large as that of the FD method, and the total CPU time for the proposed method can be reduced to about 10% of the FD method.

3.2. 2D Case
In this section, we show results from a 2D numerical modeling by the FD method and the proposed method. The 2D model is established by two isotropic media separated by a thin layer ( m, km/s), which is similar to the 1D case. As shown in Figure 3, the computational domain is 60 m × 90 m, and the thin layer is located at 72 m in direction. Displacement at two points and is observed. The sound velocity in media A is 3 km/s, and the sound velocity in media B is 4.7 km/s. Nonuniform spatial grids are used. In direction, m in the thin layer, and m in other parts. In direction, m. The external source is the same as the 1D case and is located at .
Time support for the simulation is 0.06 s. The time step is set as s and s for the FD method and the proposed method, respectively. Figure 4 shows the 2D modeling records at and for the multilayer isotropic media model. The waveform corresponding to the proposed method is in good agreement with that by the FD method, which demonstrates that the proposed method has good accuracy. Table 2 shows the comparison of CPU time between the proposed method and the FD method. Under the condition of m, the time step of the proposed method is about 3700 times as large as that of the FD method, and the total CPU time for the proposed method can be reduced to about 13% of the FD method.

(a)
(b)
4. Conclusion
In this paper, an unconditionally stable method has been proposed for solving the 1D and 2D acoustic wave equation in time domain. By applying Associated Hermit orthogonal function expansion and Galerkin temporal testing procedure, the time variable can be eliminated from the calculations and can make the proposed method unconditionally stable. The numerical results show the proposed method is efficient and the accuracy is still guaranteed.
Appendix
In order to deduce the secondorder derivation, we deduce the firstorder time derivation of the displacement first. Substituting (5) into the firstorder time derivation of (4), we obtainwhereThen we have
According to the method above, the secondorder time derivation can be expressed aswhereThen we have
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
References
 D. D. Kosloff and E. Baysal, “Forward modeling by a Fourier method,” Geophysics, vol. 56, pp. 231–241, 1982. View at: Google Scholar
 D. H. Yang, P. Tong, and X. Y. Deng, “A central difference method with low numerical dispersion for solving the scalar wave equation,” Geophysical Prospecting, vol. 60, no. 5, pp. 885–905, 2012. View at: Publisher Site  Google Scholar
 R. M. Alford, K. R. Kelly, and D. M. Boore, “Accuracy of finitedifference modeling of the acoustic wave equation,” Geophysics, vol. 39, no. 6, pp. 834–841, 1974. View at: Publisher Site  Google Scholar
 M. A. Dablain, “The application of highorder differencing to the scalar wave equation,” Geophysics, vol. 51, no. 1, pp. 54–66, 1986. View at: Publisher Site  Google Scholar
 A. Ganguli, C. M. Rappaport, D. Abramo, and S. WadiaFascetti, “Synthetic aperture imaging for flaw detection in a concrete medium,” NDT & E International, vol. 45, no. 1, pp. 79–90, 2012. View at: Publisher Site  Google Scholar
 Y. Liu and M. K. Sen, “A new timespace domain highorder finitedifference method for the acoustic wave equation,” Journal of Computational Physics, vol. 228, no. 23, pp. 8779–8806, 2009. View at: Publisher Site  Google Scholar  MathSciNet
 S. Das, W. Liao, and A. Gupta, “An efficient fourthorder low dispersive finite difference scheme for a 2D acoustic wave equation,” Journal of Computational and Applied Mathematics, vol. 258, pp. 151–167, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 W. Y. Liao, “On the dispersion, stability and accuracy of a compact higherorder finite difference scheme for 3D acoustic wave equation,” Journal of Computational and Applied Mathematics, vol. 270, pp. 571–583, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 R. K. Mohanty, “An unconditionally stable difference scheme for the onespacedimensional linear hyperbolic equation,” Applied Mathematics Letters, vol. 17, no. 1, pp. 101–105, 2004. View at: Publisher Site  Google Scholar  MathSciNet
 R. K. Mohanty and M. K. Jain, “An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation,” Numerical Methods for Partial Differential Equations, vol. 17, no. 6, pp. 684–688, 2001. View at: Publisher Site  Google Scholar  MathSciNet
 Y.S. Chung, T. K. Sarkar, B. H. Jung, and M. SalazarPalma, “An unconditionally stable scheme for the finitedifference timedomain method,” IEEE Transactions on Microwave Theory and Techniques, vol. 51, no. 3, pp. 697–704, 2003. View at: Publisher Site  Google Scholar
 Z.Y. Huang, L.H. Shi, B. Chen, and Y.H. Zhou, “A new unconditionally stable scheme for FDTD method using associated Hermite orthogonal functions,” IEEE Transactions on Antennas and Propagation, vol. 62, no. 9, pp. 4804–4809, 2014. View at: Publisher Site  Google Scholar
 H. ZhengYu, S. LiHua, Z. YingHui, and C. Bin, “AnImproved parallelinginorder solving scheme for AHFDTD method using eigenvalue transformation,” IEEE Transactions on Antennas and Propagation, vol. 63, no. 5, pp. 2135–2140, 2015. View at: Publisher Site  Google Scholar
 Z. Bi, K. Wu, C. Wu, and J. Litva, “A dispersive boundary condition for microstrip component analysis using the FDTD method,” IEEE Transactions on Microwave Theory and Techniques, vol. 40, no. 4, pp. 774–777, 1992. View at: Publisher Site  Google Scholar
 L. R. Lo Conte, R. Merletti, and G. V. Sandri, “Hermite expansions of compact support waveforms: applications to myoelectric signals,” IEEE Transactions on Biomedical Engineering, vol. 41, no. 12, pp. 1147–1159, 1994. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2015 ZhiKai Fu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.