# A GPU-Accelerated Method for 3D Nonlinear Kelvin Ship Wake Patterns Simulation
## 1 Introduction  
The focus of this research is on the innovation of highly parallel algorithms to simulate the contours of a three-dimensional free surface that appears to be stationary at the stern of a moving vessel, which are known as \"Kelvin ship waves\" [1]. Research on the kelvin wave shape has been continuously put to practical use in hull design, ship detection and environmentally friendly shipping policies [2]. [PERSON] [3], a famous naval architect, first comprehensively described the morphology and main characteristics of ship waves. Under the assumption of infinite water depth, [PERSON] [4] replaced a moving ship with a pressure disturbance point moving in a constant velocity straight line on the water surface and proposed the famous Kelvin angle of 19.47\({}^{\circ}\). In recent years, with the further study of ship wave characteristics, [PERSON] [5] noted that the wake angle will be less than the well-known Kelvin angle if the vessel speed is sufficiently large. Subsequently, various effect factors for the Kelvin wake form were discussed in plenty of papers, e.g., Froude number [6], non-axisymmetric and interference effects, shear current,surface tension, the bottom topography, submergence depth, finite water depth and viscosity [7], etc. Accordingly, the research method of ship waves has gradually shifted from the previous analytical algorithms to numerical simulation. The overwhelming majority of analytical algorithms of ship wave patterns concerns linear theories. [PERSON] [8] provided a linear solution for the problem of flow under a pressure distribution. Such ideal perturbations can also be replaced by a single submerged point source singularity [8] and submerged bodies [2]. Moreover, thin ship theory was alsoused in the study of the ship wave pattern [9]. Alongside the development of computer technology, numerical simulation methods are becoming increasing popular, and the research focus has shifted from linear problems to nonlinear problems. Nowadays, there are three numerical methods widely used to solve surface wave problems, including the boundary integral method, finite-difference method and finite-element method. In particular, [PERSON] [10] apply the boundary integral method to build a series of integro-differential equations, and the full nonlinear free surface flow problem was solved with moderate efficiencies. In more recent times, according to this method, many papers solve fully 3D nonlinear ship waves with meshes between \(60\times 20\) and \(181\times 61\)[11; 12]. And [PERSON] [6] noted that the points used along the \(x\)-direction should be more than 100 to make a sufficient standard regarding grid independence. With increasing mesh size, however, the computation time increases exponentially using only Central Processing Unit (CPU) computation power. Alongside the rapid improvement of the electronics industry, the Graphics Processing Unit (GPU) has become another method of acceleration for optimizing the execution of large numbers of threads. Currently, the powerful GPU parallel computing ability has been used to improve the studies on ocean engineering. [PERSON] [13] introduced the GPU acceleration technique into the Smoothed Particle Hydrodynamics (SPH) method to simulate complex free-surface flows, showing the high efficiency and stability of the GPU program in the SPH method. [PERSON] [14] simulated 2D dam-break flow by developing a GPU-based MPS code and achieved seven-fold speedup. As for a 3D nonlinear free surface problem, [PERSON] [15] combined the GPU acceleration technique with the boundary integral method, and [PERSON] [16] developed a GPU-accelerated high-order spectral solver. [PERSON] [17] developed the MPSGPU-SJTU solver with a GPU acceleration technique for the liquid sloshing simulation. This paper presents a parallel solution framework based on GPU for a nonlinear ship wave problem, in which almost all operations are performed in a GPU device.