A Third Order Fast Sweeping Method with Linear Computational Complexity for Eikonal Equations
 Liang Wu,
 YongTao Zhang
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Fast sweeping methods are a class of efficient iterative methods for solving steady state hyperbolic PDEs. They utilize the GaussSeidel iterations and alternating sweeping strategy to cover a family of characteristics of the hyperbolic PDEs in a certain direction simultaneously in each sweeping order. The first order fast sweeping method for solving Eikonal equations (Zhao in Math Comput 74:603–627, 2005) has linear computational complexity, namely, the computational cost is \(O(N)\) where \(N\) is the number of grid points of the computational mesh. Recently, a second order fast sweeping method with linear computational complexity was developed in Zhang et al. (SIAM J Sci Comput 33:1873–1896, 2011). The method is based on a discontinuous Galerkin (DG) finite element solver and causality indicators which guide the information flow directions of the nonlinear Eikonal equations. How to extend the method to higher order accuracy is still an open problem, due to the difficulties of solving much more complicated local nonlinear systems and calculations of local causality information. In this paper, we extend previous work and develop a third order fast sweeping method with linear computational complexity for solving Eikonal equations. A novel approach is designed for capturing the causality information in the third order DG local solver. Numerical experiments show that the method has third order accuracy and a linear computational complexity.
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 Title
 A Third Order Fast Sweeping Method with Linear Computational Complexity for Eikonal Equations
 Journal

Journal of Scientific Computing
Volume 62, Issue 1 , pp 198229
 Cover Date
 20150101
 DOI
 10.1007/s1091501498567
 Print ISSN
 08857474
 Online ISSN
 15737691
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Fast sweeping methods
 Discontinuous Galerkin methods
 High order accuracy
 Linear computational complexity
 Static Hamilton–Jacobi equations
 Eikonal equations
 65M99
 Industry Sectors
 Authors

 Liang Wu ^{(1)}
 YongTao Zhang ^{(1)}
 Author Affiliations

 1. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA
