Lax-Friedrichs sweeping scheme for static Hamilton-Jacobi equations

  • Authors:

  • Chiu Yen Kao;Stanley Osher;Jianliang Qian

  • Affiliations:

  • Department of Mathematics, University of California Los Angeles, Los Angeles, CA;Department of Mathematics, University of California Los Angeles, Los Angeles, CA;Department of Mathematics, University of California Los Angeles, Los Angeles, CA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2004

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Abstract

We propose a simple, fast sweeping method based on the Lax-Friedrichs monotone numerical Hamiltonian to approximate viscosity solutions of arbitrary static Hamilton-Jacobi equations in any number of spatial dimensions. By using the Lax-Friedrichs numerical Hamiltonian, we can easily obtain the solution at a specific 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 finite 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 efficiency and accuracy of the new approach. To our knowledge, this is the first fast numerical method based on discretizing the Hamilton-Jacobi equation directly without assuming convexity and/or homogeneity of the Hamiltonian.