| Hamilton-Jacobi equations arise in many applications such as geometrical optics, crystal growth, path planning, and seismology. Viscosity solutions of these nonlinear differential equations usually develop singularities in their derivatives. In this thesis, we will present several fast sweeping methods which are based on the Godunov Hamiltonian or the Lax-Friedrichs Hamiltonian to approximate the viscosity solution of convex or arbitrary static Hamilton-Jacobi equations in any number of spatial dimensions. We solve for the value of 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. For the sweeping methods based on Lax-Friedrichs Hamiltonian, 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 approaches. Higher order schemes are also briefly discussed. At the end, we implement Lax-Friedrichs sweeping method for optimal control problems in continuous and hybrid dynamics. |
