Characteristic fast marching method on triangular grids for the generalized eikonal equation in moving media

The governing equation of the first arrival time of a monotonically propagating front (wavefront or shock front) in an inhomogeneous moving medium is an anisotropic eikonal equation, called the generalized eikonal equation in moving media. When the ambient medium is at rest, this equation reduces to the well-known (isotropic) eikonal equation in which the characteristic direction coincides with the normal direction of the propagating front. The fast marching method is an efficient method for computing the first arrival time of a propagating front as the approximate solution of the isotropic eikonal equation. The fast marching method inherits the property that the characteristic direction coincides with the normal direction at every point on the propagating wavefront and therefore is well suited for the eikonal equation. Due to anisotropic nature, this property does not hold in the case of front propagation in a moving medium. Thus, the fast marching method cannot be directly used for the generalized eikonal equation and needs some suitable modifications. We recently proposed a characteristic fast marching method on a rectangular grid for the generalized eikonal equation (Dahiya et al., 2013) and shown numerically that this method is stable, accurate, and easy to update to second order approximations. In the present work, we generalize the method on structured triangular grids. We compare the numerical solution obtained using our method with the ray theory solution to show that the method captures accurately the viscosity solution of the generalized eikonal equation. We use the method to study some interesting geometrical features of an initially planar wavefront propagating in a medium with Taylor–Green type vortices.