Identifier: 2009-27
Author(s): Y.-T. Zhang, S. Chen, F. Li, H. Zhao, and C.-W. Shu
Title: Uniformly accurate discontinuous Galerkin fast sweeping methods for Eikonal equations
Page count: 26 pp.
Date: 2009-09-06
Abstract:
In [F. Li, C.-W. Shu, Y.-T. Zhang, H. Zhao, Journal of Computational Physics 227 (2008) 8191- 8208], we developed a fast sweeping method based on a hybrid local solver which is a combination of a discontinuous Galerkin (DG) finite element solver and a first order finite difference solver for Eikonal equations. The method has second order accuracy in the L1 norm and a very fast convergence speed, but only first order accuracy in the L∞ norm for the general cases. This is an obstacle to the design of higher order DG fast sweeping methods. In this paper, we overcome this problem by developing uniformly accurate DG fast sweeping methods for solving Eikonal equations. In order to achieve both high order accuracy and fast convergence rate (linear computational complexity), the central question is how to enforce the causality property of Eikonal equations in the compact DG local solver. We design novel causality indicators which guide the information flow directions for the DG local solver. The values of these indicators are initially provided by the first order finite difference fast sweeping method, and they are updated during iterations along with the solution. The use of causality indicators (1) allows us to compute the solution more efficiently, i.e., to only compute the solution at cells whose current causality information is consistent with the current sweeping directions, (2) is more robust than using the solution itself near singularities, such as shocks, (3) can guide the DG local solver to provide a solution for all elements of the computational mesh without switching back to the first order finite difference solver as in our previous work. We observe both a uniform second order accuracy in the L∞ norm (in smooth regions) and the fast convergence speed (linear computational complexity) in the numerical examples.
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