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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Veröffentlicht in:	Journal of scientific computing Jg. 62; H. 1; S. 198 - 229
Hauptverfasser:	Wu, Liang, Zhang, Yong-Tao
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Boston Springer US 01.01.2015 Springer Nature B.V
Schlagworte:	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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Abstract	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.
AbstractList	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. 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.
Author	Wu, Liang Zhang, Yong-Tao
Author_xml	– sequence: 1 givenname: Liang surname: Wu fullname: Wu, Liang organization: Department of Applied and Computational Mathematics and Statistics, University of Notre Dame – sequence: 2 givenname: Yong-Tao surname: Zhang fullname: Zhang, Yong-Tao email: yzhang10@nd.edu organization: Department of Applied and Computational Mathematics and Statistics, University of Notre Dame
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Cites_doi	10.1016/j.jcp.2009.05.029 10.1090/S0002-9947-1983-0690039-8 10.1137/0729053 10.1073/pnas.93.4.1591 10.1016/j.jcp.2003.11.007 10.1016/j.jcp.2008.08.016 10.1006/cviu.2000.0875 10.1016/j.jcp.2006.09.012 10.1016/j.jcp.2008.05.018 10.1073/pnas.201222998 10.1016/j.jcp.2010.07.014 10.1007/s10915-010-9345-6 10.1007/s10915-006-9124-6 10.1137/S1064827598337282 10.1007/s10915-005-9014-3 10.1137/090770291 10.1007/BF01386390 10.1137/S0036142901392742 10.1137/S0036142901396533 10.1117/12.240959 10.1090/S0025-5718-04-01678-3 10.1137/S106482759732455X 10.1109/9.412624 10.1137/S0036142997323521 10.1137/050627083 10.1137/S1064827501396798 10.4208/jcm.1003-m0016 10.1039/D0AY02022B
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Keywords	Linear computational complexity Eikonal equations Discontinuous Galerkin methods Static Hamilton–Jacobi equations High order accuracy 65M99 Fast sweeping methods
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Snippet	Fast sweeping methods are a class of efficient iterative methods for solving steady state hyperbolic PDEs. They utilize the Gauss-Seidel iterations and...
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SubjectTerms	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
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Title	A Third Order Fast Sweeping Method with Linear Computational Complexity for Eikonal Equations
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