A new method for solving the eikonal equation in the spherical computing domain

摘要

In order to overcome the problem of singularities and nonuniform grids arising when solving eikonal equation in spherical coordinate systems, a spherical Cartesian coordinate system is defined and the Hamiltonian form of the eikonal equation according to this coordinate system is given. A modified velocity function that can transform spherical coordinate system-based eikonal equation into ones based on a spherical Cartesian coordinate system is deduced by using a differential geometric method where a layered distribution of the velocity function is assumed. After comparing the results of using this approach with the traditional method of solving eikonal equation based on a spherical coordinate system, the viability of the transformation to a spherical Cartesian coordinate system based on a modified velocity function is proven. Despite the assumption of a layered distribution of the velocity function, it is also proven that the method will hold for a velocity function under any three-dimensional distribution. The new method overcomes problems present in traditional approaches and opens up a new way of solving eikonal equation in a spherical computational domain.