Sparse-Grid Implementation of Fixed-Point Fast Sweeping WENO Schemes for Eikonal Equations

作     者：Zachary M.Miksis Yong-Tao Zhang Zachary M.Miksis;Yong-Tao Zhang

作者机构：Department of Applied and Computational Mathematics and StatisticsUniversity of Notre DameNotre DameIN46556USA 

出 版 物：《Communications on Applied Mathematics and Computation》 (应用数学与计算数学学报（英文）)

年 卷 期：2024年第6卷第1期

页      面：3-29页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：supported by the NSF Grant DMS-1620108 

主　　题：Fixed-point fast sweeping methods Weighted essentially non-oscillatory(WENO)schemes Sparse grids Static Hamilton-Jacobi(H-J)equations Eikonal equations 

摘      要：Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping *** resulting iterative schemes have a fast convergence rate to steady-state ***,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local ***,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the *** multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational *** this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational ***,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)*** experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.