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A symplectic partitioned Runge-Kutta method using the eighth-order NAD operator for solving the 2D elastic wave equation

CHAOYUAN ZHANG1 ,  LI CHEN2
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1 College of Mathematics and computer, Dali University, Dali 671003, P.R. China.,
2 College of Engineering, Dali University, Dali 671003, P.R. China.,
JSE 2015, 24(3), 205–230;
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. and Chen, L., 2015. A symplectic partitioned Runge-Kutta method using the eighth-order NAD operator for solving the 2D elastic wave equation. Journal of Seismic Exploration, 24: 205- 230. In this paper, on the basis of the extended Hamiltonian system, we develop a symplectic partitioned Runge-Kutta method based on the nearly analytic discrete (NAD) operator with eighth-order accuracy for solving the 2D elastic wave equation, which is called the eighth-order NAD-SPRK method in brief. In the new method, we first employ the NAD operators with the eighth-order accuracy to discretize the high-order partial derivatives of space directions in the 2D elastic wave equation. Then the symplectic partitioned Runge-Kutta scheme with the second-order accuracy is applied to discretize the temporal high-order partial derivatives. We provide the theoretical study on the properties of the eighth-order NAD-SPRK method, such as theoretical error, stability criteria, numerical dispersion, and computational efficiency. We also compare the 2D elastic wave modeling results of this new method against those of some high-order methods. Numerical experiments show that the eighth-order NAD-SPRK method has the least numerical dispersion against the fourth-order NSPRK method, the eighth-order Lax-Wendroff correction (LWC) method, and the eighth-order staggered-grid (SG) method. Meanwhile, its computational costs and memory requirements are much less than those of the eighth-order LWC method. Against the eighth-order LWC method, comparison results indicate that the eighth-order NAD-SPRK method can provide the equivalent solutions with analytic solutions on much coarser grids. Last, we present the wave-field snapshots and wave seismograms in the homogeneous transversely isotropic medium and in the three-layer medium with a fluctuating interface for the 2D elastic wave, and the wave-field snapshots of the 2D elastic wave in the two-layer homogenous isotropic medium and in the two-layer heterogeneous medium. All these results of numerical simulations illustrate that the eighth-order NAD-SPRK method can effectively suppress the numerical dispersion caused by discretizing the wave equations when big grids are used or when models have large velocity contrasts between adjacent layers, further resulting in both saving the storage space and increasing the computational efficiency when too few sampling points per minimum wavelength are used.

Keywords
NAD operator
symplectic partitioned Runge-Kutta method
elastic wave equation
numerical dispersion
wave-field simulation
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Journal of Seismic Exploration, Electronic ISSN: 0963-0651 Print ISSN: 0963-0651, Published by AccScience Publishing
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