A High-Performance ADI–Pressure Projection Method for Solving the Navier–Stokes Equations

Resumo

The Navier–Stokes equations are fundamental for describing fluid motion, yet they remain among the most challenging problems in mathematical and computational fluid dynamics. Their strong nonlinearity, coupled with the necessity of simultaneously approximating velocity and pressure fields, leads to a highly complex and computationally demanding system. In particular, mixed formulations introduce compatibility requirements that add further difficulty to ensuring stable and accurate solutions. Moreover, the consistent enforcement of mass conservation remains a delicate aspect in numerical schemes. To address these challenges, this work proposes a simple and efficient numerical strategy that combines an Alternating Direction Implicit (ADI) method with a Pressure Projection technique. The ADI method efficiently handles the velocity computation, while the Pressure Projection ensures the enforcement of the divergence-free condition, preserving mass conservation naturally. For enhanced computational performance, the velocity field is solved in CPU using OpenMP parallelization in the ADI steps, while GPU acceleration using CUDA is employed for the solution of the pressure equation. The proposed methodology is validated through convergence studies and comparisons with benchmark solutions from the literature, applied to classical CFD test cases. The results demonstrate the accuracy, efficiency, and robustness of the approach, highlighting its potential for scalable and reliable simulations in computational fluid dynamics.