A GPU IMPLEMENTATION OF THE TSUNAMI EQUATION
DOI:
https://doi.org/10.37943/13SCQO3041Keywords:
Tsunami equation, finite difference scheme, numerical methods, parallel algorithm, CUDAAbstract
In this paper, we consider numerical simulation and GPU (graphics processing unit) computing for the two-dimensional non-linear tsunami equation, which is a fundamental equation of tsunami propagation in shallow water areas. Tsunamis are highly destructive natural disasters that have a significant impact on coastal regions. These events are typically caused by undersea earthquakes, volcanic eruptions, landslides, and possibly an asteroid impact. To solve numerically, firstly we discretized these equations in a rectangular domain and then transformed the partial differential equations into semi-implicit finite difference schemes. The spatial and time derivatives are approximated by using the second-order centered differences following the Crank-Nicolson method and the calculation method is based on the Jacobi method; the computation is performed using the C++ programming language; and the visualization of numerical results is performed by Matlab 2021. The initial condition was given as a Gaussian, and the basin profile has been approximated by a hyperbolic tangent. To accelerate the sequential algorithm, a parallel computation algorithm is developed using CUDA (Compute Unified Device Architecture) technology. CUDA technology has long been used for the numerical solution of partial differential equations (PDEs). It uses the parallel computing capabilities of graphics processing units (GPUs) to speed up the PDE solution. By taking advantage of the GPU’s massive parallelism, CUDA technology can significantly speed up PDE computations, making it an effective tool for scientific computing in a variety of fields. The performance of the parallel implementation is tested by comparing the computation time between the sequential (CPU) solver and CUDA implementations for various mesh sizes. The comparison shows that our parallel implementation gives significant acceleration in the implementation of CUDA.
References
Klockner, A., Warburton, T., Bridge, J., & Hesthaven, J.S. (2009). Nodal discontinuous Galerkin methods on graphics processors, Journal of Computational Physics, 228(21),7863-7882. https://doi.org/10.1016/j.jcp.2009.06.041
Bell, N., & Garland, M. (2008). Efficient sparse matrix-vector multiplication on CUDA (Vol. 2, No. 5). Nvidia Technical Report NVR-2008-004, Nvidia Corporation.
Elsen, E., LeGresley, P., & Darve, E. (2008) Large calculation of the flow over a hypersonic vehicle using a GPU, Journal of Computational Physics, 227, 10148-10161. https://doi.org/10.1016/j.jcp.2008.08.023
Gidra, H., Haque, I., Kumar, N.P., Sargurunathan, M., Gaur, M.S., Laxmi, V., ... & Singh, V. (2011, September). Parallelizing TUNAMI-N1 Using GPGPU. In 2011 IEEE International Conference on High Performance Computing and Communications (pp. 845-850). IEEE. https://doi.org/10.1109/HPCC.2011.120
Goto, C., Ogawa, Y., Shuto, N., & Imamura, F. (1997). Numerical method of tsunami simulation with the leap-frog scheme. IOC Manuals and Guides, 35, 130.
Titov, V. V., & Synolakis, C. E. (1995). Modeling of breaking and nonbreaking long-wave evolution and runup using VTCS-2. Journal of Waterway, Port, Coastal, and Ocean Engineering, 121(6), 308-316. https://doi.org/10.1061/(ASCE)0733-950X(1995)121:6(308)
Wang, X., & Liu, P.L.F. (2006). An analysis of 2004 Sumatra earthquake fault plane mechanisms and Indian Ocean tsunami. Journal of Hydraulic Research, 44(2), 147-154. https://doi.org/10.1080/00221686.2006.9521671
Amouzgar, R., Liang, Q., Clarke, P.J., Yasuda, T., & Mase, H. (2016). Computationally efficient tsunami modeling on graphics processing units (GPUs). International Journal of Offshore and Polar Engineering, 26(02), 154-160. https://doi.org/10.17736/ijope.2016.ak10
Arnoldy, A., & Adytia, D. (2019, July). Performance of Staggered Grid Implementation of 2D Shallow Water Equations using CUDA Architecture. In 2019 12th International Conference on Information & Communication Technology and System (ICTS) (pp. 286-290). IEEE. https://doi.org/10.1109/ICTS.2019.8850930
Asunción, M., Castro, M.J., Mantas, J.M., & Ortega, S. (2016). Numerical simulation of tsunamis generated by landslides on multiple GPUs. Advances in Engineering Software, 99, 59-72. https://doi.org/10.1016/j.advengsoft.2016.05.005
Asunción, M., Mantas, J.M., & Castro, M.J. (2010). Programming CUDA-based GPUs to simulate two-layer shallow water flows. In Euro-Par 2010-Parallel Processing: 16th International Euro-Par Conference, Ischia, Italy, August 31-September 3, 2010, Proceedings, Part II 16 (pp. 353-364). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-15291-7_32
Khrapov, S.S., & Khoperskov, A.V. (2020). Application of Graphics Processing Units for self consistent modelling of shallow water dynamics and sediment transport. Lobachevskii Journal of Mathematics, 41, 1475-1484. https://doi.org/10.1134/S1995080220080089
Boubekeur, M., Benkhaldoun, F., & Seaid, M. (2017). GPU accelerated finite volume methods for three-dimensional shallow water flows. In Finite Volumes for Complex Applications VIII-Hyperbolic, Elliptic and Parabolic Problems: FVCA 8, Lille, France, June 2017 8 (pp. 137-144). Springer International Publishing. https://doi.org/10.1007/978-3-319-57394-6_15
Asunción, M., & Castro, M. J. (2017). Simulation of tsunamis generated by landslides using adaptive mesh refinement on GPU. Journal of Computational Physics, 345, 91-110. https://doi.org/10.1016/j.jcp.2017.05.016
Satria, M.T., Huang, B., Hsieh, T.J., Chang, Y.L., & Liang, W.Y. (2012). GPU acceleration of tsunami propagation model. IEEE Journal of Selected Topics in Applied Earth Observations and remote Sensing, 5(3), 1014-1023. https://doi.org/10.1109/JSTARS.2012.2199468
Nagasu, K., Sano, K., Kono, F., & Nakasato, N. (2017). FPGA-based tsunami simulation: Performance comparison with GPUs, and roofline model for scalability analysis. Journal of Parallel and Distributed Computing, 106, 153-169. https://doi.org/10.1016/j.jpdc.2016.12.015
Parna, P., Meyer, K., & Falconer, R. (2018). GPU driven finite difference WENO scheme for real time solution of the shallow water equations. Computers & Fluids, 161, 107-120. https://doi.org/10.1016/j.compuid.2017.11.012
Zhai, J., Liu, W., & Yuan, L. (2016). Solving two-phase shallow granular flow equations with a well balanced NOC scheme on multiple GPUs. Computers & Fluids, 134, 90-110. https://doi.org/10.1016/j.compuid.2016.04.0
Imamura F. & Yalcine A.C. (2006). Tsunami Modeling Manual, 58 pages. Retrieved from http://www.tsunami.civil.tohoku.ac.jp/hokusai3/J/projects/manual-ver-3.1.pdf
NVIDIA TURING GPU ARCHITECTURE. Graphics Reinvented. Retrieved from https://www.nvidia.com/content/dam/en-zz/Solutions/design-visualization/technologies/turing-architecture/NVIDIATuring-Architecture-Whitepaper.pdf
Altybay, A., Ruzhansky, M., & Tokmagambetov, N. (2020). A parallel hybrid implementation of the 2D acoustic wave equation. International Journal of Nonlinear Sciences and Numerical Simulation, 21(7-8), 821-827. https://doi.org/10.1515/ijnsns-2019-0227
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Articles are open access under the Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors who publish a manuscript in this journal agree to the following terms:
- The authors reserve the right to authorship of their work and transfer to the journal the right of first publication under the terms of the Creative Commons Attribution License, which allows others to freely distribute the published work with a mandatory link to the the original work and the first publication of the work in this journal.
- Authors have the right to conclude independent additional agreements that relate to the non-exclusive distribution of the work in the form in which it was published by this journal (for example, to post the work in the electronic repository of the institution or publish as part of a monograph), providing the link to the first publication of the work in this journal.
- Other terms stated in the Copyright Agreement.