In this research work, we apply a numerical scheme based on the first-order time integration approach combined with the modifications of the meshless approximation for solving the conservative Allen–Cahn–Navier–Stokes equations. More precisely, we first utilize a first-order time discretization for the Navier–Stokes equations and the time-splitting technique of order one for the dynamics of the phase-field variable. Besides, we use the local interpolation based on the Matérn radial function for spatial discretization. We should solve a Poisson equation with the proper boundary conditions to have the divergence-free property during the numerical algorithm. Accordingly, the applied numerical procedure could not give a stable and accurate solution. Instead, we solve a regularization system in a discrete form. To prevent the instability of the numerical solution concerning the convection term, a biharmonic term with a small coefficient based on the high-order hyperviscosity formulation has been added, which has been approximated by a scalable interpolation based on the combination of polyharmonic spline with polynomials (known as the PHS+poly). The obtained full-discrete problem is solved using the biconjugate gradient stabilized method considering a proper preconditioner. We investigate the potency of the numerical scheme by presenting some simulations via uniform, hexagonal, and quasi-uniform nodes on rectangular and irregular domains. Besides, we have compared the proposed meshless method with the standard finite element method due to the used CPU time.

中文翻译：

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用于求解不可压缩两相流体流动的局部 RBF 插值及其修改：保守的 Allen–Cahn–Navier–Stokes 系统


在这项研究工作中，我们应用基于一阶时间积分方法的数值方案结合无网格近似的修改来求解保守的 Allen-Cahn-Navier-Stokes 方程。更准确地说，我们首先利用纳维-斯托克斯方程的一阶时间离散化和相场变量动力学的一阶时间分割技术。此外，我们使用基于Matérn径向函数的局部插值进行空间离散化。为了使数值算法具有无散性，需要求解具有适当边界条件的泊松方程。因此，所应用的数值程序无法给出稳定且准确的解。相反，我们以离散形式求解正则化系统。为了防止对流项数值解的不稳定，添加了基于高阶超粘性公式的小系数双调和项，该双调和项通过基于多调和样条与多项式组合的可扩展插值来近似（称为小灵通+聚）。考虑到适当的预处理器，使用双共轭梯度稳定方法解决了所获得的全离散问题。我们通过在矩形和不规则域上的均匀、六边形和准均匀节点进行一些模拟来研究数值方案的效力。此外，由于所使用的 CPU 时间，我们将所提出的无网格方法与标准有限元方法进行了比较。