An Unconditionally Stable Method for Solving the Acoustic Wave Equation

Abstract

An unconditionally stable method for solving the time-domain acoustic wave equation using Associated Hermit orthogonal functions is proposed. The second-order 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 Courant-Friedrichs-Levy (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.

Academic Editor:Mitsuhiro Okayasu

National Key Laboratory on Electromagnetic Environmental Effects and Electro-optical Engineering, PLA University of Science and Technology, Nanjing 210007, China

Received 16 April 2015; Revised 21 July 2015; Accepted 26 July 2015; 11 August 2015

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.

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 time-marching algorithm, the time step must be limited by the Courant-Friedrichs-Levy (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 alternating-direction 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 first-order Maxwell's equation.

In this paper, we extended the unconditionally stable method using AH functions to solve the second-order acoustic wave equation. Firstly, the second-order 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 media [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the sound velocity, [figure omitted; refer to PDF] is the wave displacement, and [figure omitted; refer to PDF] is an external source. In order to truncate the computational domain, we choose Mur's first-order absorbing boundary condition (ABC) for infinite problem from [14]. At the boundary points [figure omitted; refer to PDF] or [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF]

Consider a set of modified AH orthogonal basis functions given by [12] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the transformed time variable, [figure omitted; refer to PDF] is the time-translating parameter, and [figure omitted; refer to PDF] is the time-scaling parameter. By selecting proper [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , the modified AH orthogonal basis functions can be used to approximate the causal displacement [figure omitted; refer to PDF] by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] th expansion coefficient; [figure omitted; refer to PDF] is selected according to the bandwidth of the analyzed signal [15].

With the time derivation property of AH functions [15] [figure omitted; refer to PDF]

We can express the first-order time derivation and the second-order time derivation of the displacement [figure omitted; refer to PDF] as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] represents the order in the AH domain. The deduction can be found in the Appendix.

Substituting (4) and (7) into (1), we obtain [figure omitted; refer to PDF]

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] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . We can partition the spatial domain [figure omitted; refer to PDF] into [figure omitted; refer to PDF] . Then the spatial discrete form of (9) at point [figure omitted; refer to PDF] can be written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] th order coefficients and the [figure omitted; refer to PDF] th order coefficients to the [figure omitted; refer to PDF] th order coefficients, we can rewrite (10) in a matrix form: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] in a unit matrix. Consider [figure omitted; refer to PDF]

Here, [figure omitted; refer to PDF] is a tridiagonal coefficient matrix. [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] to [figure omitted; refer to PDF] : [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a tridiagonal matrix with submatrix elements. [figure omitted; refer to PDF] is a combination of all [figure omitted; refer to PDF] in the whole computation domain. [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] ) [figure omitted; refer to PDF]

Using the average techniques, we have [figure omitted; refer to PDF]

Substituting (16) into (15), we have [figure omitted; refer to PDF]

Applying (17) into all orders in the AH domain, we can obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Using the technique above, the absorbing boundary condition at point [figure omitted; refer to PDF] can be expressed as [figure omitted; refer to PDF]

Introducing the boundary condition (18) and (20) into (11), (13) can be replaced by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is an adaptive coefficient matrix including absorbing boundary condition and [figure omitted; refer to PDF] is a coefficient matrix corresponding to [figure omitted; refer to PDF] .

We can use the lower-upper (LU) decomposition method to decompose [figure omitted; refer to PDF] and use the back-substitution method to calculate the coefficients of the displacement for all points. The displacement [figure omitted; refer to PDF] 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 by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is divided into [figure omitted; refer to PDF] grid.

Substituting (4) and (7) into (22) and rearranging it, we can obtain a matrix form equation which is similar to (11): [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the spatial grid size in [figure omitted; refer to PDF] direction and [figure omitted; refer to PDF] is the spatial grid size in [figure omitted; refer to PDF] direction. In (23), we can find each point has a relationship with its adjacent four points, so matrix [figure omitted; refer to PDF] is a five-diagonal coefficient matrix in 2D case.

For the absorbing boundary condition, the boundary condition can be obtained by using (15) to (20). But the corner-point needs special treatment and the spatial grid size is adjusted to [figure omitted; refer to PDF] at corner-point. For the four corner-points, Mur's absorbing boundary condition can be expressed as [figure omitted; refer to PDF]

Introducing Mur's absorbing boundary condition, the coefficient matrix [figure omitted; refer to PDF] can be adjusted to [figure omitted; refer to PDF] . Using the lower-upper (LU) decomposition method and the back-substitution method, the coefficients of all points can be calculated. And the displacement [figure omitted; refer to PDF] 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 ( [figure omitted; refer to PDF] m, [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] at two points [figure omitted; refer to PDF] (38 m) and [figure omitted; refer to PDF] (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 [figure omitted; refer to PDF] m; in other parts [figure omitted; refer to PDF] m.

Figure 1: Configuration of a thin layer ( [figure omitted; refer to PDF] m) embedded in two isotropic media.

[figure omitted; refer to PDF]

The external source is located at [figure omitted; refer to PDF] m and its expression is [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] Hz. Time support for the simulation is 0.16 s. To expand [figure omitted; refer to PDF] in AH domain, we select [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Due to the limitation of the CFL condition, the time step is set as [figure omitted; refer to PDF] s for the FD method under the condition of [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] s is good enough. The absolute error [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF] for comparison of the two methods. [figure omitted; refer to PDF] is the reference data obtained from the FD method with [figure omitted; refer to PDF] m, and [figure omitted; refer to PDF] is the numerical result obtained from the FD method ( [figure omitted; refer to PDF] m) and the proposed method ( [figure omitted; refer to PDF] m). The numerical results at [figure omitted; refer to PDF] and their errors are shown in Figure 2(a). Figure 2(b) is the numerical results at [figure omitted; refer to PDF] . 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 ( [figure omitted; refer to PDF] m) is much lower than that of the FD method ( [figure omitted; refer to PDF] m) at both [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Figure 2: The numerical results and their errors at (a) [figure omitted; refer to PDF] and (b) [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Table 1 shows the comparison of CPU time between the proposed method and the FD method. Under the condition of [figure omitted; refer to PDF] 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.

Table 1: The comparison of CPU time.

	[figure omitted; refer to PDF] = 0.01 m
[figure omitted; refer to PDF]	CPU time
FD method	[figure omitted; refer to PDF] s	3.198 s
Proposed method	[figure omitted; refer to PDF] s	0.3127 s

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 ( [figure omitted; refer to PDF] m, [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] direction. Displacement [figure omitted; refer to PDF] at two points [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] direction, [figure omitted; refer to PDF] m in the thin layer, and [figure omitted; refer to PDF] m in other parts. In [figure omitted; refer to PDF] direction, [figure omitted; refer to PDF] m. The external source is the same as the 1D case and is located at [figure omitted; refer to PDF] .

Figure 3: Configuration of a thin layer ( [figure omitted; refer to PDF] m) embedded in two isotropic media.

[figure omitted; refer to PDF]

Time support for the simulation is 0.06 s. The time step is set as [figure omitted; refer to PDF] s and [figure omitted; refer to PDF] s for the FD method and the proposed method, respectively. Figure 4 shows the 2D modeling records at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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.

Table 2: The comparison of CPU time.

	[figure omitted; refer to PDF] m
[figure omitted; refer to PDF]	CPU time
FD method	[figure omitted; refer to PDF] s	237.9171 s
Proposed method	[figure omitted; refer to PDF] s	30.8813 s

Figure 4: The numerical results at (a) [figure omitted; refer to PDF] and (b) [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Appendix

Appendix

In order to deduce the second-order derivation, we deduce the first-order time derivation of the displacement first. Substituting (5) into the first-order time derivation of (4), we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Then we have [figure omitted; refer to PDF]

According to the method above, the second-order time derivation can be expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Then we have [figure omitted; refer to PDF]