A Third Order Fast Sweeping Method with Linear Computational Complexity for Eikonal Equations

Fast sweeping methods are a class of efficient iterative methods for solving steady state hyperbolic PDEs. They utilize the Gauss-Seidel iterations and alternating sweeping strategy to cover a family of characteristics of the hyperbolic PDEs in a certain direction simultaneously in each sweeping ord...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 62; no. 1; pp. 198 - 229
Main Authors: Wu, Liang, Zhang, Yong-Tao
Format: Journal Article
Language:English
Published: Boston Springer US 01.01.2015
Springer Nature B.V
Subjects:
Accuracy
Algorithms
Causality
Complexity
Computation
Computational grids
Computational Mathematics and Numerical Analysis
Computing costs
Design
Eikonal equation
Hyperbolic differential equations
Information flow
Iterative methods
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical models
Mathematics
Mathematics and Statistics
Methods
Nonlinear systems
Numerical analysis
Partial differential equations
Solvers
Sweeping
Theoretical
Linear computational complexity
Eikonal equations
Discontinuous Galerkin methods
Static Hamilton–Jacobi equations
High order accuracy
65M99
Fast sweeping methods
ISSN:0885-7474, 1573-7691
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Summary:Fast sweeping methods are a class of efficient iterative methods for solving steady state hyperbolic PDEs. They utilize the Gauss-Seidel 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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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-014-9856-7