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A Runge-Kutta method with using eighth-order nearly-analytic spatial discretization operator for solving a 2D acoustic wave equation

CHAOYUAN ZHANG1,2 ,  XIAO LI2 ,  XIAO MA2 ,  GUOJIE SONG3
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College of Mathematics and Computer, Dali University, Dali 671003, P.R. China. zcy_km@163.com,
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China.,
Department of Computer Science and Technology, Tsinghua University, Beijing 100084, P.R. China.,
JSE 2014, 23(3), 279–302;
Submitted: 9 June 2025 | Revised: 9 June 2025 | Accepted: 9 June 2025 | Published: 9 June 2025
© 2025 by the Authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

Zhang, C., Li, X., Ma, X. and Song, G., 2014. A Runge-Kutta method with using eighth-order nearly-analytic spatial discretization operator for solving a 2D acoustic wave equation. Journal of Seismic Exploration, 23: 279-302. In this paper, we develop an eighth-order NAD-RK method for solving a 2D acoustic wave equation. The new method uses an eighth-order nearly-analytic discretization (NAD) operator to approximate the high-order spatial derivatives in the wave equation. The wavefield displacements and their gradients are used simultaneously. And we apply a third-order Runge-Kutta (RK) method to solve the semi-discrete ordinary differential equations (ODEs) with respect to time. Thus this method has third-order accuracy in time and can achieve eighth-order accuracy in space. Theoretical properties including stability and errors are analyzed for the eighth-order NAD-RK method in detail. Meanwhile, the numerical dispersion relationship for this method is investigated and the numerical dispersion is tested in our study. The study shows that the eighth-order NAD-RK method has the smallest numerical dispersion and the weakest numerical dispersion anisotropy compared with the eighth-order Lax-Wendroff correction (LWC) method and the eighth-order staggered-grid (SG) method. The computational efficiency of the eighth-order NAD-RK method is also tested. The results show that the eighth-order NAD-RK method needs less computational time and requires less memory than the eighth-order LWC and SG methods. Finally, the eighth-order NAD-RK method is used to simulate acoustic wavefields for two heterogeneous layered models. The simulation results further demonstrate that the eighth-order NAD-RK method can provide high-order accuracy for the complex heterogeneous models and is effective to suppress the numerical dispersion caused by discretizing the acoustic wave equation when too coarse grids are used or strong discontinuities exist in the medium. Thus, the eighth-order NAD-RK method can be potentially used in seismic tomography and large-scale wave propagation problems.

Keywords
RK method
numerical dispersion
acoustic wave equation
NAD operator
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Journal of Seismic Exploration, Electronic ISSN: 0963-0651 Print ISSN: 0963-0651, Published by AccScience Publishing
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