Robust and efficient multilevel-ILU preconditioning of hybrid Newton–GMRES for incompressible Navier–Stokes equations,International Journal for Numerical Methods in Fluids

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Robust and efficient multilevel-ILU preconditioning of hybrid Newton–GMRES for incompressible Navier–Stokes equations
International Journal for Numerical Methods in Fluids ( IF 1.7 ) Pub Date : 2021-08-12 , DOI: 10.1002/fld.5039
Qiao Chen ₁ , Xiangmin Jiao ₁ , Oliver Yang ₁

Affiliation

We introduce a robust and efficient preconditioner for a hybrid Newton–GMRES method for solving the nonlinear systems arising from incompressible Navier–Stokes equations. When the Reynolds number is relatively high, these systems often involve millions of degrees of freedom (DOFs), and the nonlinear systems are difficult to converge, partially due to the strong asymmetry of the system and the saddle-point structure. In this work, we propose to alleviate these issues by leveraging a multilevel ILU preconditioner called HILUCSI, which is particularly effective for saddle-point problems and can enable robust and rapid convergence of the inner iterations in Newton–GMRES. We further use Picard iterations with the Oseen systems to hot-start Newton–GMRES to achieve global convergence, also preconditioned using HILUCSI. To further improve efficiency and robustness, we use the Oseen operators as physics-based sparsifiers when building preconditioners for Newton iterations and introduce adaptive refactorization and iterative refinement in HILUCSI. We refer to the resulting preconditioned hybrid Newton–GMRES as HILUNG. We demonstrate the effectiveness of HILUNG by solving the standard 2D driven-cavity problem with Re 5000 and a 3D flow-over-cylinder problem with low viscosity. We compare HILUNG with some state-of-the-art customized preconditioners for INS, including two variants of augmented Lagrangian preconditioners and two physics-based preconditioners, as well as some general-purpose approximate-factorization techniques. Our comparison shows that HILUNG is much more robust for solving high-Re problems and it is also more efficient in both memory and runtime for moderate-Re problems.

中文翻译：

用于不可压缩 Navier-Stokes 方程的混合 Newton-GMRES 稳健且高效的多级 ILU 预处理

我们为混合 Newton-GMRES 方法引入了一种稳健有效的预处理器，用于求解由不可压缩 Navier-Stokes 方程引起的非线性系统。当雷诺数较高时，这些系统往往涉及数百万个自由度（DOF），非线性系统难以收敛，部分原因是系统的强不对称性和鞍点结构。在这项工作中，我们建议通过利用称为HILUCSI的多级 ILU 预处理器来缓解这些问题，这对于鞍点问题特别有效，并且可以实现 Newton-GMRES 中内部迭代的稳健和快速收敛。我们进一步使用皮卡德迭代与 Oseen 系统热启动 Newton-GMRES 以实现全局收敛，也使用 HILUCSI 进行预处理。为了进一步提高效率和鲁棒性，我们在为 Newton 迭代构建预处理器时使用 Oseen 算子作为基于物理的稀疏器，并在 HILUCSI 中引入自适应重构和迭代细化。我们将由此产生的预处理混合 Newton-GMRES 称为HILUNG. 我们通过使用 Re 5000 解决标准 2D 驱动腔问题和具有低粘度的 3D 流过圆柱体问题来证明 HILUNG 的有效性。我们将 HILUNG 与一些最先进的 INS 定制预处理器进行了比较，包括两种增强拉格朗日预处理器的变体和两种基于物理的预处理器，以及一些通用的近似分解技术。我们的比较表明，HILUNG 在解决高 Re 问题方面更加稳健，并且在解决中等 Re 问题时在内存和运行时方面也更加高效。

更新日期：2021-08-12

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