On the stability and efficiency of high-order parallel algorithms for 3D wave problems
DOI: https://doi.org/10.3846/mma.2025.23819Abstract
In this work, we investigate the stability conditions for four new high-order ADI type schemes proposed to solve 3D wave equations with a non-constant sound speed coefficient. This analysis is mainly based on the spectral method, therefore a basic benchmark problem is formulated with a constant sound speed coefficient. For a case of general non-constant coefficient the stability analysis is done by using the energy method. Our main conclusion states that the selected ADI type schemes use different factorization operators (mainly due to the need to approximate the artificial boundary conditions on the split time levels), but the general structure of the stability factors are similar for all schemes and thus the obtained CFL conditions are also very similar.
The second goal is to compare the accuracy and efficiency of the selected ADI solvers. This analysis also includes parallel versions of these schemes. Two schemes are selected as the most effective and accurate.
Keywords:
ADI discrete schemes, 3D wave problem, stability, parallel algorithmsHow to Cite
Share
License
Copyright (c) 2025 The Author(s). Published by Vilnius Gediminas Technical University.
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
S. Britt, E. Turkel and S. Tsynkov. A high order compact time/space finite difference scheme for the wave equation with variable speed of sound. J. Sci.Comput., 76(2):777–811, 2018. https://doi.org/10.1007/s10915-017-0639-9
D.W. Deng. Numerical simulation of the coupled sine-Gordon equations via a linearized and decoupled compact ADI method. Numer. Funct. Anal. Optim., 40(9):1053–1079, 2019. https://doi.org/10.1080/01630563.2019.1596951
W. Hundsdorfer and J. Verwer. Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Series in Computational Mathematics, volume 33. Springer, Berlin, Heidelberg, New York, Tokyo, 2003. https://doi.org/10.1007/978-3-662-09017-6
Y. Jiang and Y. Ge. An explicit high-order compact finite difference scheme for the three-dimensional acoustic wave equation with variable speed of sound. International Journal Of Computer Math., 100(2):321–341, 2023. https://doi.org/10.1080/00207160.2022.2118524
K. Li and W. Liao. An efficient and high accuracy finite-difference scheme for the acoustic wave equation in 3d heterogeneous media. J. Comp. Science, 40:101063, 2020. https://doi.org/10.1016/j.jocs.2019.101063
K. Li, W. Liao and Y. Lin. A compact high order alternating direction implicit method for three-dimensional acoustic wave equation with variable coefficient. Journal of Computational and Applied Mathematics, 361(1):113–129, 2019. https://doi.org/10.1016/j.cam.2019.04.013
W. Liao. On the dispersion, stability and accuracy of a compact higher-order finite difference scheme for 3d acoustic wave equation. J. Comput. Appl. Math., 270:571–583, 2014. https://doi.org/10.1016/j.cam.2013.08.024
B. Otero, O. Rojas, F. Moya and J.E. Castillo. Alternating direction implicit time integrations for finite difference acoustic wave propagation: parallelization and convergence. Computers and Fluids, 205:104584, 2020. https://doi.org/10.1016/j.compfluid.2020.104584
A.A. Samarskii. The Theory of Difference Schemes. Marcel Dekker, New York, 2001. https://doi.org/10.1201/9780203908518
Gerald Teschl. Mathematical Methods in Quantum Mechanics, volume 157. American Mathematical Soc., New York, 2014.
W. Zhang. A new family of fourth-order locally one-dimensional schemes for the 3D elastic wave equation. J. Comput.Appl. Math., 348:246–260, 2019. https://doi.org/10.1016/j.cam.2018.08.056
W. Zhang and J. Jiang. A new family of fourth-order locally one-dimensional schemes for the three-dimensional wave equation. J. Computational and Applied Math., 311:130–147, 2017. https://doi.org/10.1016/j.cam.2016.07.020
A. Zlotnik and R. Čiegis. On higher-order compact ADI schemes for the variable coefficient wave equation. Applied Mathematics and Computation, 412(2):126565, 2022. https://doi.org/10.1016/j.amc.2021.126565
A. Zlotnik and R. Čiegis. On construction and properties of compact 4th order finite-difference schemes for the variable coefficient wave equation. J. Sci Comput, 95(3):2–35, 2023. https://doi.org/10.1007/s10915-023-02127-3
A. Zlotnik and O. Kireeva. On compact 4th order finite-difference schemes for the wave equation. Math. Model. Anal., 26(3):479–502, 2021. https://doi.org/10.3846/mma.2021.13770
View article in other formats
CrossMark check
Published
Issue
Section
Copyright
Copyright (c) 2025 The Author(s). Published by Vilnius Gediminas Technical University.
License
This work is licensed under a Creative Commons Attribution 4.0 International License.