An explicit high-order compact finite difference scheme for the three-dimensional acoustic wave equation with variable speed of sound

Abstract

In this paper, the correction for the remainder of the truncation error of the second-order central difference scheme is employed to discretize temporal derivative, while the fourth-order Padé schemes are directly used to compute spatial derivatives, an explicit high-order compact finite difference scheme to solve the three-dimensional acoustic wave equation with variable speed of sound is proposed. This new scheme has the fourth-order accuracy in both temporal and spatial directions. It has high computational efficiency since the Thomas algorithm is employed to solve three tridiagonal linear systems formed by the Padé schemes on the (n)th time step and then an explicit time advancement process is conducted for the $(n+1)$th time step. The stability and convergence condition of the proposed scheme are proved. We extend the proposed method to solve problems with nonlinear source terms and systems. Numerical experiments are conducted to demonstrate theoretical analysis results of the proposed scheme.

Keywords:

• Acoustic wave equation
• three-dimensional
• high-order compact scheme
• explicit
• stability and convergence

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35L05
• 65M06
• 65M12

Acknowledgments

We would like to thank the editors and the anonymous referees whose constructive comments and suggestions are helpful to improve the quality of this paper. The authors are grateful to Prof. Zhi-zhong Sun for his valuable discussions and suggestions which lead to improvement of this paper. We also express our thanks to Prof. Wenyuan Liao, who kindly provides the program codes for us to conveniently make comparisons with the numerical solutions in the literature.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by National Natural Science Foundation of China [grant numbers 12161067, 11772165, 11961054, 11902170], National Natural Science Foundation of Ningxia [grant numbers 2022AAC02023, 2020AAC03059], the Key Research and Development Program of Ningxia [grant number 2018BEE03007], National Youth Top-notch Talent Support Program of Ningxia, and the First Class Discipline Construction Project in Ningxia Universities: Mathematics.