A Hybrid Finite Difference WENO-ZQ Fast Sweeping Method for Static Hamilton–Jacobi Equations

• Published in: Journal of scientific computing Vol. 83; no. 3; p. 54
• Main Authors: Ren, Yupeng, Xiong, Tao, Qiu, Jianxian
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2020; Springer Nature B.V
• Subjects: Algorithms; Approximation; Boundary value problems; Computational Mathematics and Numerical Analysis; Conservation laws; Finite difference method; Hamilton-Jacobi equation; Iterative methods; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Parameter sensitivity; Stencils; Sweeping; Theoretical; WENO; Hybrid scheme; Finite difference; 65M60; Static Hamilton–Jacobi equation; Fast sweeping method; 35L65
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-020-01228-7

In this paper, we propose to combine a new fifth order finite difference weighted essentially non-oscillatory (WENO) scheme with high order fast sweeping methods, for directly solving static Hamilton–Jacobi equations. This is motivated by the work in Xiong et al. (J Sci Comput 45(1–3):514–536, 2010), where a fifth order fast sweeping method base on the classical finite difference WENO scheme is developed. Numerical results in Xiong et al. (2010) show that the iterative numbers of the scheme for some cases are very sensitive to the parameter ϵ , which is used to avoid the denominator to be 0 in the nonlinear weights. Here we propose to use the new fifth order finite difference WENO-ZQ scheme, which was recently developed in Zhu and Qiu (J Comput Phys 318:110–121, 2016), to alleviate this problem. Besides, to save computational cost from WENO reconstructions, a hybrid finite difference linear and WENO scheme is used, which works more robustly. Numerical experiments will be performed to demonstrate the good performance of the new proposed approach.