An improved mixed Lagrangian–Eulerian (IMLE) method for modelling incompressible Navier–Stokes flows with CUDA programming on multi-GPUs

•The IMLE method is proposed to solve the incompressible Navier–Stokes equations.•Multiple GPUs are adopted to accelerate the computation.•A data decomposition strategy is proposed to achieve higher speedup ratio.•The speedup ratio can up to 70x for adopting four GPU cards. In this study, a GPU-accelerated improved mixed Lagrangian–Eulerian (IMLE) method is proposed to solve the three-dimensional incompressible Navier–Stokes equations. To improve the prediction accuracy, the proposed IMLE method approximates the total derivative term in Lagragian sense, and the spatial derivative terms are approximated on Eulerian coordinates. Transfer of data from Lagrangian particles to data on Eulerian grids is accurately carried out by adopting moving least squares (MLS) interpolation method. The velocity-pressure decoupling issue is overcome by adopting pressure-free projection method in which the pressure field is calculated by solving a pressure Poisson equation (PPE). It is noted that the MLS interpolation is time consuming since this procedure belongs to a pointwise scheme in which a local matrix equation shall be solved on each grid point. In addition, the discretized PPE forms a large sparse matrix and it is computationally intensive to solve by using the conjugate gradient (CG) method. Therefore, we are aimed to resort to CUDA- and OpenMP-programming means to accelerate the computation. In this study, the performance of the multiple GPUs code can reach up to 27 times faster with respect to multi-threads CPU performance.