Characteristic Fast Marching Method for Monotonically Propagating Fronts in a Moving Medium

Abstract

The fast marching method is computationally efficient in approximating the viscosity solution of the eikonal equation in the case of unidirectional wavefront propagation through a medium at rest. The main assumption of this method is that the front propagates only in its normal direction, which is the case when the medium of propagation is at rest. In many real-time applications, the medium may be occupied with a moving fluid. In such cases, the governing equation is a generalized (anisotropic) eikonal equation. The main assumption of the fast marching method may not hold in this case, since the front may propagate in both the tangential and the normal direction. This leads to instability in the fast marching method due to violation of the upwind criterion. In this work, we develop a fast marching method for the generalized eikonal equation, called the characteristic fast marching method, where the upwind criterion is achieved using the characteristic direction of the propagating wavefront at each grid point. We suitably modify the narrow band algorithm of the fast marching method so that the anisotropic nature of the medium is incorporated in the method. We compare the numerical results obtained from our method with the solution obtained using the ray theory (geometrical optics theory) to show that the method accurately captures the viscosity solution of the generalized eikonal equation. We apply the method to study the propagation of a wavefront in a medium with a cavity and also study the merging of two wavefronts from different sources. The method can easily be generalized to higher order approximations. We develop a method with second order finite difference approximation and study the rate of convergence numerically.