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arXiv · 2512.18422

A pressure-projection formulation in a least-squares meshfree method for the incompressible Navier-Stokes equations using a staggered-variable arrangement

Takeharu Matsuda·Satoshi Ii
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Abstract

Incompressible flow solvers based on strong-form meshfree methods represent arbitrary geometries without the need for a global mesh system. However, their local evaluations make it difficult to satisfy incompressibility at the discrete level. Moreover, the collocated arrangement of velocity and pressure variables tends to induce a zero-energy mode, leading to decoupling between two variables. In projection-based approaches, a spatial discretization scheme based on a conventional node-based moving least-squares method for the pressure causes inconsistency between discrete operators on both sides of the Poisson equation. Thus, a solenoidal velocity field cannot be ensured numerically. In this study, a numerical method for the incompressible Navier-Stokes equations is developed by introducing a local primal-dual grid into the mesh-constrained discrete point method, enabling consistent discrete operators. The constructed virtual dual cell is defined solely from local node connectivity, and thus the method retains its meshfree nature. To consistently couple velocity and pressure variables under primal-dual arrangement, a numerical technique called time evolution converting, which evaluates interface quantities by considering the temporal evolution of flow, is used to update the velocity at cell interfaces. For numerical validation, a linear acoustic equation is solved to confirm the effectiveness of the staggered-variable arrangement based on the local primal-dual grid. Then, the incompressible Navier-Stokes equations are solved, and the proposed method is demonstrated to reduce the discrete velocity divergence to a level determined by the residual of the pressure Poisson equation, achieve the expected spatial convergence order, and accurately reproduce flow features over a wide range of Reynolds numbers.

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Takeharu Matsuda, Satoshi Ii. 2026-09-17. A pressure-projection formulation in a least-squares meshfree method for the incompressible Navier-Stokes equations using a staggered-variable arrangement. https://arxiv.org/abs/2512.18422

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