发表机构
School of Mathematics and Statistics, Huang Huai University; College of Science and Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing University of Aeronautics and Astronautics; Department of Applied and Computational Mathematics and Statistics, University of Notre Dame（河南工程学院数学与统计学院; 南京航空航天大学理学院及工信部航空器数学建模与高性能计算重点实验室; 圣母大学应用与计算数学和统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对经典WENO-JS局部求解器导致快速扫描方法难以全收敛的问题，提出冻结权重技术，开发了全收敛五阶不动点快速扫描方法，数值实验验证其能稳定收敛至舍入误差。
AI 中文摘要
带加权本质无振荡（WENO）局部求解器的不动点快速扫描方法是一类求解双曲守恒律稳态解的高效高阶精度数值方法。然而，使用经典WENO-JS局部求解器时，高阶不动点快速扫描格式的迭代残差往往难以降至舍入误差水平。为实现快速扫描方法的全收敛，基于不等长子模板的非传统WENO局部求解器的不动点快速扫描方法被设计出来。但基于不等长子模板的WENO格式比经典WENO-JS格式更复杂，且通常计算成本更高。本文回归经典WENO-JS局部求解器，开发了一种新的全收敛五阶不动点快速扫描方法，用于求解双曲守恒律的稳态问题。基于近期关于解间断附近非线性加权过程的研究，我们应用冻结权重技术，以避免WENO-JS局部求解器中非线性权重的不必要调整，即在快速扫描迭代中残差序列稳定后冻结非线性权重。与现有冻结权重工作不同，我们设计了一种简单且稳健的方法来判断迭代残差的稳定性，并确定在快速扫描方法中冻结非线性权重的迭代步。大量针对具有挑战性的二维稳态问题的数值实验表明，与之前带五阶WENO-JS局部求解器的快速扫描方法不同，所提出的新格式始终能将迭代残差降至舍入误差水平，实现全收敛。
英文摘要
The fixed-point fast sweeping methods with weighted essentially non-oscillatory (WENO) local solvers are a class of efficient and high-order accuracy numerical methods for solving steady-state solutions of hyperbolic conservation laws. However, with the classical WENO-JS local solver, the iteration residue of high-order fixed-point fast sweeping scheme often has difficulty to settle down to round-off errors. To achieve the full convergence in a fast sweeping method, the fixed-point fast sweeping methods with non-traditional WENO local solvers based on unequal-sized substencils were designed. However, the WENO schemes based on unequal-sized stencils are more complex and in general more expensive in computational costs than the classical WENO-JS schemes. In this paper, we go back to the classical WENO-JS local solver and develop a new fully convergent fifth-order fixed-point fast sweeping method for solving steady-state problems of hyperbolic conservation laws. Based on recent studies on the nonlinear weighting process around discontinuities of solution, we apply the technique of frozen weights for avoiding unnecessary adjustment of nonlinear weights in the WENO-JS local solver, which freezes the nonlinear weights once the residual sequence in the fast sweeping iterations has stabilized. Different from the existing work on frozen weights, we design a simple and robust approach to judge stabilization of iteration residues and determine the iteration step when the nonlinear weights are frozen in the fast sweeping method. Extensive numerical experiments on a wide range of challenging two-dimensional steady-state problems demonstrate that, unlike the previous fast sweeping method with the fifth-order WENO-JS local solver, the proposed new scheme consistently drives the iteration residues to round-off errors and achieves the full convergence.
