A viscosity solutions approach to shape-from-shading
SIAM Journal on Numerical Analysis
A Discontinuous Galerkin Finite Element Method for Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
Weighted ENO Schemes for Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
Computer Vision and Image Understanding
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
SIAM Journal on Numerical Analysis
High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes
SIAM Journal on Scientific Computing
Ordered Upwind Methods for Static Hamilton--Jacobi Equations: Theory and Algorithms
SIAM Journal on Numerical Analysis
Lax-Friedrichs sweeping scheme for static Hamilton-Jacobi equations
Journal of Computational Physics
Fast Sweeping Methods for Static Hamilton--Jacobi Equations
SIAM Journal on Numerical Analysis
Short note: O(N) implementation of the fast marching algorithm
Journal of Computational Physics
High Order Fast Sweeping Methods for Static Hamilton---Jacobi Equations
Journal of Scientific Computing
A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations
Journal of Computational Physics
A Fast Sweeping Method for Static Convex Hamilton-Jacobi Equations
Journal of Scientific Computing
Fast Sweeping Methods for Eikonal Equations on Triangular Meshes
SIAM Journal on Numerical Analysis
A second order discontinuous Galerkin fast sweeping method for Eikonal equations
Journal of Computational Physics
Journal of Computational Physics
Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws
Journal of Computational Physics
Journal of Scientific Computing
A high order moving boundary treatment for compressible inviscid flows
Journal of Computational Physics
Fast Two-scale Methods for Eikonal Equations
SIAM Journal on Scientific Computing
A uniformly second order fast sweeping method for eikonal equations
Journal of Computational Physics
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In [F. Li, C.-W. Shu, Y.-T. Zhang, H. Zhao, J. Comput. Phys., 227 (2008) pp. 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 $L^1$ norm and a very fast convergence speed, but only first order accuracy in the $L^\infty$ 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. 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. We observe both a uniform second order accuracy in the $L^\infty$ norm (in smooth regions) and the fast convergence speed (linear computational complexity) in the numerical examples.