The nodal integral methods (NIMs) have found widespread use in the nuclear industry for neutron transport problems due to their high accuracy. However, despite considerable development, these methods have limited acceptability among the fluid flow community. One major drawback of these methods is the lack of robust and efficient nonlinear solvers for the algebraic equations resulting from discretization. Since its inception, several modifications have been made to make NIMs more agile, efficient, and accurate. Modified nodal integral method (MNIM) and modified MNIM (M²NIM) are the two most recent and efficient versions of the NIM for fluid flow problems. M²NIM modifies the MNIM by replacing the current time convective velocity with the previous time convective velocity, leading to faster convergence albeit with reduced accuracy. This work proposes a preconditioned Jacobian-free Newton–Krylov approach for solving the Navier–Stokes equation using MNIM. The Krylov solvers do not generally work well without an appropriate preconditioner. Therefore, M²NIM is used here as a preconditioner to accelerate the solution of MNIM. Due to pressure–velocity coupling in the Navier–Stokes equation, developing a quality preconditioner for these equations needs significant effort. The momentum equation is solved using the time-splitting alternate direction implicit method. The velocities obtained from the solution are then used to solve the pressure Poisson equation. The computational results for the Navier–Stokes equation are presented to underscore the advantages of the developed algorithm.

中文翻译：

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使用节点积分法对纳维-斯托克斯方程的无雅可比牛顿-克雷洛夫求解器进行基于物理的预处理

节点积分法（NIM）由于其高精度而在核工业中广泛用于解决中子输运问题。然而，尽管取得了相当大的发展，这些方法在流体流动领域的可接受性仍然有限。这些方法的一个主要缺点是缺乏稳健且高效的非线性求解器来求解离散化产生的代数方程。自成立以来，NIM 进行了多次修改，使 NIM 更加敏捷、高效和准确。改进的节点积分法 (MNIM) 和改进的 MNIM (M ² NIM) 是用于流体流动问题的 NIM 的两个最新且有效的版本。M ² NIM 通过用前一时间的对流速度替换当前时间的对流速度来修改 MNIM，从而导致更快的收敛，但精度降低。这项工作提出了一种使用 MNIM 求解纳维-斯托克斯方程的预处理无雅可比牛顿-克雷洛夫方法。如果没有适当的预处理器，Krylov 求解器通常无法正常工作。因此，这里使用M ² NIM作为预条件子来加速MNIM的求解。由于纳维-斯托克斯方程中的压力-速度耦合，为这些方程开发高质量的预处理器需要付出巨大的努力。使用时分交替方向隐式方法求解动量方程。然后使用从解中获得的速度来求解压力泊松方程。纳维-斯托克斯方程的计算结果强调了所开发算法的优势。