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.
Numerical simulation of the acoustic wave equation has been widely used in many areas, such as geophysics, exploration, and ultrasonic detection [
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.
For simplicity, we consider the 1D acoustic wave equation with an external source in isotropic media
Consider a set of modified AH orthogonal basis functions given by [
With the time derivation property of AH functions [
We can express the firstorder time derivation and the secondorder time derivation of the displacement
Substituting (
Based on the orthogonal property of the AH functions, we can obtain equations of the AH expansion coefficients using the temporal Galerkin testing procedure [
In (
Here,
Substitute (
Using the average techniques, we have
Substituting (
Applying (
Using the technique above, the absorbing boundary condition at point
Introducing the boundary condition (
We can use the lowerupper (LU) decomposition method to decompose
Considering the need for practical application, we extend the method to the 2D acoustic wave equation, which is given by
Substituting (
For the absorbing boundary condition, the boundary condition can be obtained by using (
Introducing Mur’s absorbing boundary condition, the coefficient matrix
In order to validate the proposed method, a 1D structure constituted by two isotropic media separated by a thin layer (
Configuration of a thin layer (
The external source is located at
The numerical results and their errors at (a)
Table
The comparison of CPU time.
 


CPU time  
FD method 

3.198 s 
Proposed method 

0.3127 s 
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 (
Configuration of a thin layer (
Time support for the simulation is 0.06 s. The time step is set as
The comparison of CPU time.
 


CPU time  
FD method 

237.9171 s 
Proposed method 

30.8813 s 
The numerical results at (a)
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.
In order to deduce the secondorder derivation, we deduce the firstorder time derivation of the displacement first. Substituting (
According to the method above, the secondorder time derivation can be expressed as
The authors declare that there is no conflict of interests regarding the publication of this paper.