High-accuracy numerical solver for the Eikonal equation - Guo, Jin Long (Nanyang Technological University, 2021)
An introduction to the Eikonal equation and its applications in Physics as well as the Earth Sciences is provided, followed by two derivations of the Eikonal equation. Proofs of existence and uniqueness to the Eikonal equation initial value problem are provided. The Fast Sweeping Method (FSM) is then introduced in detail and run on two dimensional Cartesian coordinates. The numerical results are compared again analytic results and its accuracy is evaluated using percentage relative error. Relative error was found to be reduced by decreasing grid size. The FSM algorithm was also run on three dimensional spherical coordinates as well as Cartesian coordinates using parallel computing techniques. FSM was also implemented on two dimensional Cartesian domain triangulated with acute triangles. In all cases, it was shown that the FSM convergerd quickly after minimal iterations and approximated the actual solution to Eikonal equations to which the analytical solution is known to a high degree of accuracy. Furthermore, it was found that parallel computing techniques greatly improved computing efficiency by reducing convergence time. Some limitations of this project include the type of grid used to discretise the computational domain as well as the type of source used to initialise the Eikonal equation. It is recommended that future projects experiment with adaptive meshes and also possibly linear sources (instead of point sources) which often arise in the field of Seismology.