# A GPU-Accelerated Method for 3D Nonlinear Kelvin Ship Wake Patterns Simulation
## 6 Conclusions  
The numerical simulation of ship waves is important for practical ocean engineering. This paper proposes a highly paralleled numerical scheme for simulating three-dimensional (3-D) nonlinear Kelvin ship waves effectively, including a numerical model for nonlinear ship waves, a banded preconditioner JFNK method and a GPU-based parallel computing framework. Numerical simulations show that the proposed GPU solver can save GPU memory and obtain high efficiency significantly. This highly paralleled numerical scheme provides an opportunity for the further study of the nonlinear Kelvin ship waves on a large scale.  
1. The bandwidth has an effect on the running memory and runtime of the GPU solver. Based on the mesh size, the value of the most appropriate bandwidth is around \(\frac{M}{3}\times(N+1)\); more than 66% GPU memory can be saved.
2. The GPU solver can obtain an accurate numerical solution. The mean square error of the GPU solver results and CPU solver results is \(MSE=9.37\times 10^{-8}\), which is acceptable.
3. By designing the GPU parallel computing framework, the computation of ship wave simulation is accelerated up to 20 times.  
Although a highly paralleled numerical scheme for nonlinear ship waves is proposed in this paper, some assumptions are still made in the construction of the numerical model,  
Figure 13: Wake pattern of real ship waves and the GPU solver computational results. The picture of a real speedboat wake pattern came from the internet [[https://www.quanjing.com](https://www.quanjing.com)]([https://www.quanjing.com](https://www.quanjing.com)), accessed on 1 September 2023; the picture of a real fishing ship wake came from [[https://www.shutterstock.com](https://www.shutterstock.com)]([https://www.shutterstock.com](https://www.shutterstock.com)), accessed on 1 September 2023; the picture of a real large vessel wake came from the internet [[https://blogs.worldbank.org](https://blogs.worldbank.org)]([https://blogs.worldbank.org](https://blogs.worldbank.org)), accessed on 6 September 2023.  
such as infinite water depth and the steady motion of a ship on calm water. It is of great significance to improve simulation results by further exploring the influence of finite water depth, tangential flow and unsteady ship motion on nonlinear ship waves.  
Validation, X.S., M.C. and J.D.; investigation, X.S.; writing--original draft preparation, X.S. and M.C.; writing--review and editing, X.S. and M.C.; supervision, X.S. All authors have read and agreed to the published version of the manuscript.  
The work was supported by the National Key R&D Program of China (No. 2022 YFB4300803, 2022 YFB4301402), the Ministry of Industry and Information Technology of the People's Republic of China (No. CBG3N21-3-3), and the National Science Foundation of Liaoning Province, China (No. 2022-MS-159). The authors would like to express sincere thanks for their support.  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.