Another look at velocity exten sions in the level set method

The level set method [S. Osher and J. A. Set hian, J. Comput. Phys., 79 (1988), pp. 12-49] has become a widely used numerical method for moving interfaces; e.g., see the many examples in [J. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Material Science, Cambridge University Press, London, 1996; S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer Verlag, Heidelberg, 2002]. For many applications, the velocity of the interface is known only on the interface, while the level set method requires information about the interface speed at least in a neighborhood of grid points near the interface. To address this issue, velocity extensions are us ed to map the velocity information on the interface into the rest of the computational domain [D. Adalsteinsson and J. Sethian, J. Comput. Phys., 118 (1995), pp. 269-277]. This allows the level set method to proceed. The velocity extension method presented in [D. Adalsteinsson and J. Sethian, J. Comput. Phys., 118 (1995), pp. 269-277] uses the fast marching method [J. Sethian, Proc. N atl. Acad. Sci., 93 (1996), pp. 1591- 1595; J. Sethian, SIAM Rev., 41 (1999), pp. 199-235], and is only a first-order approximation for the velocity field near the interface. Furthermore, it can lead to unexpected behavior in some cases. This is primarily due to the strictly local solution for the characterist ics of the flow near the interface. In this paper, we look more closely at the characteristics near the interface, and then present a modified velocity extension method, also based on the fast marching meth od, which handles the characteristcs more accurately. In turn, this new method will lead to some interesting new possibilities for the fast marching method.