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

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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.

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Acknowledgments

We thank helpful discussions with Chi-Wang Shu and Hongkai Zhao on this project

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Authors and Affiliations

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

    Liang Wu & Yong-Tao Zhang

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  1. Liang Wu
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  2. Yong-Tao Zhang
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Correspondence to Yong-Tao Zhang.

Appendix: Detailed Formulae of the \(6 \times 6\) Jacobian Matrix in Solving the Local Nonlinear System

Appendix: Detailed Formulae of the \(6 \times 6\) Jacobian Matrix in Solving the Local Nonlinear System

$$\begin{aligned} J(1,1)&= \frac{\partial f_1}{\partial \phi }=l_j(\alpha _l-\alpha _r)+h_i(\alpha _b-\alpha _t) \\ J(1,2)&= \frac{\partial f_1}{\partial u}=8k~u_{ij}+l_j(-\alpha _l-\alpha _r) \\ J(1,3)&= \frac{\partial f_1}{\partial v}=\frac{8}{k}~v_{ij}+h_i(-\alpha _b-\alpha _t) \\ J(1,4)&= \frac{\partial f_1}{\partial a}=\frac{32k}{3}~a_{ij}+l_j(\alpha _l-\alpha _r)+\frac{1}{3}h_i(\alpha _b-\alpha _t) \\ J(1,5)&= \frac{\partial f_1}{\partial b}=\frac{32}{3k}~b_{ij}+\frac{1}{3}l_j(\alpha _l-\alpha _r)+h_i(\alpha _b-\alpha _t) \\ J(1,6)&= \frac{\partial f_1}{\partial c}=(\frac{8k}{3}+\frac{8}{3k})~c_{ij} \end{aligned}$$
$$\begin{aligned} J(2,1)&= \frac{\partial f_2}{\partial \phi }=l_j(-\alpha _l-\alpha _r) \\ J(2,2)&= \frac{\partial f_2}{\partial u}=\frac{16k}{3}~a_{ij}+l_j(\alpha _l-\alpha _r)+\frac{1}{3}h_i(\alpha _b-\alpha _t) \\ J(2,3)&= \frac{\partial f_2}{\partial v}=\frac{8}{3k}~c_{ij} \\ J(2,4)&= \frac{\partial f_2}{\partial a}=\frac{16k}{3}~u_{ij}+l_j(-\alpha _l-\alpha _r) \\ J(2,5)&= \frac{\partial f_2}{\partial b}=\frac{1}{3}l_j(-\alpha _l-\alpha _r) \\ J(2,6)&= \frac{\partial f_2}{\partial c}=\frac{8}{3k}~v_{ij}+\frac{1}{3}h_i(-\alpha _b-\alpha _t) \end{aligned}$$
$$\begin{aligned} J(3,1)&= \frac{\partial f_3}{\partial \phi }=h_i(-\alpha _b-\alpha _t) \\ J(3,2)&= \frac{\partial f_3}{\partial u}=\frac{8k}{3}~c_{ij} \\ J(3,3)&= \frac{\partial f_3}{\partial v}=\frac{16}{3k}~b_{ij} + \frac{1}{3}l_j(\alpha _l-\alpha _r)+h_i(\alpha _b-\alpha _t) \\ J(3,4)&= \frac{\partial f_3}{\partial a}=\frac{1}{3}h_i(-\alpha _b-\alpha _t) \\ J(3,5)&= \frac{\partial f_3}{\partial b}=\frac{16}{3k}v_{ij} + h_i(-\alpha _b-\alpha _t) \\ J(3,6)&= \frac{\partial f_3}{\partial c}=\frac{8k}{3}~u_{ij}+ \frac{1}{3}l_j(-\alpha _l-\alpha _r) \end{aligned}$$
$$\begin{aligned} J(4,1)&= \frac{\partial f_4}{\partial \phi }= l_j(\alpha _l-\alpha _r)+\frac{1}{3}h_i(\alpha _b-\alpha _t) \\ J(4,2)&= \frac{\partial f_4}{\partial u}=\frac{8k}{3}~u_{ij}+ l_j(-\alpha _l-\alpha _r) \\ J(4,3)&= \frac{\partial f_4}{\partial v}=\frac{8}{3k}~v_{ij}+\frac{1}{3}h_i(-\alpha _b-\alpha _t) \\ J(4,4)&= \frac{\partial f_4}{\partial a}=\frac{32k}{5}~a_{ij}+ l_j(\alpha _l-\alpha _r)+\frac{1}{5}h_i(\alpha _b-\alpha _t) \\ J(4,5)&= \frac{\partial f_4}{\partial b}= \frac{32}{9k}~b_{ij}+ \frac{1}{3}\big ( l_j(\alpha _l-\alpha _r)+h_i(\alpha _b-\alpha _t) \big ) \\ J(4,6)&= \frac{\partial f_4}{\partial c}= (\frac{8k}{9}+\frac{8}{5k})~c_{ij} \end{aligned}$$
$$\begin{aligned} J(5,1)&= \frac{\partial f_5}{\partial \phi }=\frac{1}{3}l_j(\alpha _l-\alpha _r)+h_i(\alpha _b-\alpha _t) \\ J(5,2)&= \frac{\partial f_5}{\partial u}=\frac{8k}{3}~u_{ij}+\frac{1}{3}l_j(-\alpha _l-\alpha _r) \\ J(5,3)&= \frac{\partial f_5}{\partial v}=\frac{8}{3k}~v_{ij}+h_i(-\alpha _b-\alpha _t) \\ J(5,4)&= \frac{\partial f_5}{\partial a}=\frac{32k}{9}~a_{ij}+\frac{1}{3}\big ( l_j(\alpha _l-\alpha _r)+h_i(\alpha _b-\alpha _t) \big ) \\ J(5,5)&= \frac{\partial f_5}{\partial b}=\frac{32}{5k}~b_{ij}+ \frac{1}{5}l_j(\alpha _l-\alpha _r)+h_i(\alpha _b-\alpha _t) \\ J(5,6)&= \frac{\partial f_5}{\partial c}= (\frac{8k}{5}+\frac{8}{9k})~c_{ij} \end{aligned}$$
$$\begin{aligned} J(6,1)&= \frac{\partial f_6}{\partial \phi }=0 \\ J(6,2)&= \frac{\partial f_6}{\partial u}=\frac{1}{3}h_i(-\alpha _b-\alpha _t) \\ J(6,3)&= \frac{\partial f_6}{\partial v}=\frac{1}{3}l_j(-\alpha _l-\alpha _r) \\ J(6,4)&= \frac{\partial f_6}{\partial a}=\frac{16k}{9}~c_{ij} \\ J(6,5)&= \frac{\partial f_6}{\partial b}=\frac{16}{9k}~c_{ij} \\ J(6,6)&= \frac{\partial f_6}{\partial c}=\frac{16k}{9}~a_{ij}+\frac{16}{9k}~b_{ij}+\frac{1}{3}\big ( l_j(\alpha _l-\alpha _r)+h_i(\alpha _b-\alpha _t) \big ) \end{aligned}$$

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Wu, L., Zhang, YT. A Third Order Fast Sweeping Method with Linear Computational Complexity for Eikonal Equations. J Sci Comput 62, 198–229 (2015). https://doi.org/10.1007/s10915-014-9856-7

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