High Order Absolutely Convergent Fast Sweeping Methods with Multi-resolution WENO Local Solvers for Eikonal and Factored Eikonal Equations

Fast sweeping methods are a class of efficient iterative methods developed in the literature to solve steady-state solutions of hyperbolic partial differential equations (PDEs). In Zhang et al. (J Sci Comput 29:25-56, 2006) and Xiong et al. (J Sci Comput 45:514-536, 2010), high order accuracy fast sweeping schemes based on classical weighted essentially non-oscillatory (WENO) local solvers were developed for solving static Hamilton-Jacobi equations. However, since high order classical WENO methods (e.g., fifth order and above) often suffer from difficulties in their convergence to steady-state solutions, iteration residues of high order fast sweeping schemes with these local solvers may hang at a level far above round-off errors even after many iterations. This issue makes it difficult to determine the convergence criterion for the high order fast sweeping methods and challenging to apply the methods to complex problems. Motivated by the recent work on absolutely convergent fast sweeping method for steady-state solutions of hyperbolic conservation laws in Li et al. (J Comput Phys 443:110516, 2021), in this paper we develop high order fast sweeping methods with multi-resolution WENO local solvers for solving Eikonal equations, an important class of static Hamilton-Jacobi equations. Based on such kind of multi-resolution WENO local solvers with unequal-sized sub-stencils, iteration residues of the designed high order fast sweeping methods can settle down to round-off errors and achieve the absolute convergence. In order to obtain high order accuracy for problems with singular source-point, we apply the factored Eikonal approach developed in the literature and solve the resulting factored Eikonal equations by the new high order WENO fast sweeping methods. Extensive numerical experiments are performed to show the accuracy, computational efficiency, and advantages of the new high order fast sweeping schemes for solving static Hamilton-Jacobi equations.