An enhanced localized method of approximate particular solutions (LMAPS) for solving two-dimensional incompressible Navier–Stokes equations

This paper presents an enhanced localized method of approximate particular solutions (LMAPS) for solving two-dimensional, transient, incompressible Navier–Stokes equations in primitive variables. Building upon the original LMAPS framework, we introduce an Hermite interpolation within local stencils, significantly refining the solution accuracy for high-gradient flows, particularly in regions with sharp flow variations. The interpolation now adapts to local grid density and curvature, providing a more accurate approximation of both velocity and pressure fields. The proposed method still solves the Navier-Stokes system by incorporating local matrices into a global sparse system but with improved resolution and fewer iterations required. Numerical experiments, including the driven cavity problem and flow around a cylinder, demonstrate that the modified LMAPS achieves higher accuracy, improved stability, and better convergence, especially at high Reynolds numbers, compared to the standard approach.