Journal of Scientific Computing

, Volume 81, Issue 3, pp 2484–2502 | Cite as

Improved Characteristic Fast Marching Method for the Generalized Eikonal Equation in a Moving Medium

  • Myong-Song HoEmail author
  • Ju-Hyok Ri
  • Sin-Bom Kim


The first arrival time of a monotonically propagating front (wavefront or shock front) in an inhomogeneous moving medium is governed by an anisotropic eikonal equation (generalized eikonal equation in a moving medium). The characteristic fast marching method (CFMM developed by Dahiya et al.) is a fast and accurate method for such anisotropic eikonal equation. Unfortunately, this method fails for some discretization parameters, and one can hardly choose the discretization parameters on which the CFMM succeeds in the case when the magnitude of the external velocity is comparable to the speed of propagation of the wavefront. In this work we develop an improved characteristic fast marching method which overcomes the limitation of the CFMM. We discuss in detail why the CFMM fails for some discretization parameters and how the limitation can be overcome. We propose a procedure (called the difference correction procedure) so as to capture the viscosity solution of the anisotropic eikonal equation independently of the discretization parameters. We use our method to obtain the first arrival time of an initially planar wavefront propagating in a medium with Taylor–Green type vortices, and also apply our method to a sinusoidal waveguide example that has been widely used in the seismic community as a model problem. We compare the numerical solutions obtained using our method with the solutions obtained using the ray theory to show that our method captures the viscosity solution of the generalized eikonal equation in a moving medium accurately.


Fast marching method Anisotropic front propagation Ray tracing method Viscosity solution 



The authors would like to thank the referees for their valuable comments and suggestions, which helped the authors to improve the article significantly.


This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsKim Il Sung UniversityPyongyangDemocratic People’s Republic of Korea

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