We present an exponentially convergent semi-implicit meshless algorithm for the solution of Navier-Stokes equations in complex domains. The algorithm discretizes partial derivatives at scattered points using radial basis functions as interpolants. Higher-order polynomials are appended to polyharmonic splines (PHS-RBF) and a collocation method is used to derive the interpolation coefficients. The interpolating kernels are then differentiated and the partial-differential equations are satisfied by collocation at the scattered points. The PHS-RBF interpolation is shown to be exponentially convergent with discretization errors decreasing as a high power of a representative distance between points. We present here a semi-implicit algorithm for time-dependent and steady state fluid flows in complex domains. At each time step, several iterations are performed to converge the momentum and continuity equations. A Poisson equation for pressure corrections is formulated by imposing divergence free condition on the iterated velocity field. At each time step, the momentum and pressure correction equations are repeatedly solved until the velocities and pressure converge to a pre-specified tolerance. We have demonstrated the convergence and discretization accuracy of the algorithm for two model problems and simulated three other complex problems. In all cases, the algorithm is stable for Courant numbers in excess of ten. The algorithm has the potential to accurately and efficiently solve many fluid flow and heat transfer problems in complex domains. An open source code Meshless Multi-Physics Software (MeMPhyS) is available for interested users of the algorithm.

中文翻译：

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复杂几何中不可压缩流动的一种半隐式无网格方法

我们提出了一种指数收敛的半隐式无网格算法，用于求解复杂域中的 Navier-Stokes 方程。该算法使用径向基函数作为插值在散点处离散偏导数。将高阶多项式附加到多谐样条 (PHS-RBF) 并使用搭配方法来推导插值系数。然后对插值核进行微分，并通过在散点处的搭配来满足偏微分方程。PHS-RBF 插值显示为指数收敛，离散化误差随着点之间的代表性距离的高次幂而降低。我们在此提出了一种用于复杂域中瞬态和稳态流体流动的半隐式算法。在每个时间步，执行多次迭代以收敛动量和连续性方程。压力修正的泊松方程是通过在迭代速度场上施加无发散条件来制定的。在每个时间步长，动量和压力校正方程都会重复求解，直到速度和压力收敛到预先指定的容差。我们已经证明了算法对两个模型问题的收敛性和离散化精度，并模拟了其他三个复杂问题。在所有情况下，该算法对于超过 10 的 Courant 数都是稳定的。该算法有可能准确有效地解决复杂领域中的许多流体流动和传热问题。一个开源代码无网格多物理软件 (MeMPhyS) 可供对该算法感兴趣的用户使用。