Background: Partial differential equations (PDEs) constitute the mathematical backbone of fluid dynamics, governing mass, momentum, and energy transport in continuum media. Numerical approaches to solving these equations have grown in sophistication alongside advances in high-performance computing.
Objective: This study comparatively evaluates three dominant numerical schemes—Finite Difference Methods (FDM), Finite Element Methods (FEM), and Finite Volume Methods (FVM)—for solving incompressible Navier–Stokes equations across benchmark fluid flow configurations.
Methods: Benchmark simulations including lid-driven cavity, backward-facing step, Poiseuille channel flow, and turbulent pipe flow were executed at varying Reynolds numbers. Stability, L2 error norms, and CPU runtime were measured systematically.
Results: FDM demonstrated the lowest computational cost for laminar regimes; FEM provided superior accuracy in geometrically complex domains; FVM offered the best balance of stability and efficiency for turbulent cases. L2 error norms ranged from 9.6 × 10⁻⁴ to 4.3 × 10⁻³.
Conclusion: Method selection should be guided by flow complexity, geometric constraints, and available computational resources. Hybrid and adaptive schemes are identified as a promising frontier for next-generation CFD solvers.