by Chiu Yen Kao, Stanley Osher, Jianliang Qian
Journal of Computational Physics
ftp://ftp.math.ucla.edu/pub/camreport/cam03-38.ps.gz
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Abstract:
We propose a simple, fast sweeping method based on the Lax-Friedrichs monotone numerical Hamiltonian to approximate the viscosity solution of arbitrary static Hamilton-Jacobi equations in any number of spatial dimensions. By using the Lax-Friedrichs numerical Hamiltonian, we can easily solve for the value of a specic grid point in terms of its neighbors, so that a Gauss-Seidel type nonlinear iterative method can be utilized. Furthermore, by incorporating a group-wise causality principle into the Gauss-Seidel iteration by following a nite group of characteristics, we have an easy-to-implement, sweeping-type, and fast convergent numerical method. However, unlike other methods based on the Godunov numerical Hamiltonian, some computational boundary conditions are needed in the implementation. We give a simple recipe which enforces a version of discrete min-max principle. Some convergence analysis is done for the one-dimensional eikonal equation. Extensive 2-D and 3-D numerical examples illustrate the eciency and accuracy of the new approach. To our knowledge, this is the rst fast numerical method based on discretizing the Hamilton-Jacobi equation directly without assuming convexity and/or homogeneity of the Hamiltonian. 1
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