# American Institute of Mathematical Sciences

October  2015, 8(5): 871-880. doi: 10.3934/dcdss.2015.8.871

## Distance function and extension in normal direction for implicitly defined interfaces

 1 Department of Mathematics and Descriptive Geometry, Slovak University of Technology, Bratislava, Slovak Republic, Slovak Republic, Slovak Republic

Received  January 2014 Revised  June 2014 Published  July 2015

In this paper we present a novel application of extrapolation procedure for three popular numerical algorithms to compute the distance function for an interface that is given only implicitly. The methods include the fast marching method [8], the fast sweeping method [10] and the linearization method [10]. The extrapolation procedure removes the necessity of a special initialization procedure for the grid nodes next to the interface that is used so far with the methods, thus it represents a natural extension of these methods. The extrapolation procedure can be used also for an extension of a function that is defined only locally on the interface in the direction given by the gradient of distance function [2].
Citation: Peter Frolkovič, Karol Mikula, Jozef Urbán. Distance function and extension in normal direction for implicitly defined interfaces. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 871-880. doi: 10.3934/dcdss.2015.8.871
##### References:
 [1] D. Adalsteinsson and J. Sethian, The fast construction of extension velocities in level set methods, J. Comput. Phys., 148 (1999), 2-22. doi: 10.1006/jcph.1998.6090. [2] T. D. Aslam, A partial differential equation approach to multidimensional extrapolation, J. Comput. Phys., 193 (2004), 349-355. doi: 10.1016/j.jcp.2003.08.001. [3] S. Fomel, Traveltime Computation with the Linearized Eikonal Equation, Technical report, SEP 94, 1997. [4] P. Frolkovič, Flux-based level set method for extrapolation along characteristics using immersed interface formulation, In P. Struk, editor, Magia, Slovak University of Technology, Bratislava, (2010), 15-26. [5] S. Hysing and S. Turek, The Eikonal equation: numerical efficiency vs. algorithmic complexity on quadrilateral grids, In Proceedings of Algoritmy, 2005, (2005), 22-31. [6] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer, 2003. doi: 10.1007/b98879. [7] E. Rouy and A. Tourin, A viscosity solutions approach to shape-from-shading, SIAM J. Num. Anal., 29 (1992), 867-884. doi: 10.1137/0729053. [8] J. Sethian, A fast marching level set method for monotonically advancing fronts, Proc. Nat. Acad. Sci., 93 (1996), 1591-1595. doi: 10.1073/pnas.93.4.1591. [9] J. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999. [10] H. Zhao, A fast sweeping method for eikonal equations, Math. Comput., 74 (2005), 603-627. doi: 10.1090/S0025-5718-04-01678-3. [11] H. Zhao, T. Chan, B. Merriman and S. Osher, A variational level set approach to multiphase motion, J. Comput. Phys., 127 (1996), 179-195. doi: 10.1006/jcph.1996.0167.

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##### References:
 [1] D. Adalsteinsson and J. Sethian, The fast construction of extension velocities in level set methods, J. Comput. Phys., 148 (1999), 2-22. doi: 10.1006/jcph.1998.6090. [2] T. D. Aslam, A partial differential equation approach to multidimensional extrapolation, J. Comput. Phys., 193 (2004), 349-355. doi: 10.1016/j.jcp.2003.08.001. [3] S. Fomel, Traveltime Computation with the Linearized Eikonal Equation, Technical report, SEP 94, 1997. [4] P. Frolkovič, Flux-based level set method for extrapolation along characteristics using immersed interface formulation, In P. Struk, editor, Magia, Slovak University of Technology, Bratislava, (2010), 15-26. [5] S. Hysing and S. Turek, The Eikonal equation: numerical efficiency vs. algorithmic complexity on quadrilateral grids, In Proceedings of Algoritmy, 2005, (2005), 22-31. [6] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer, 2003. doi: 10.1007/b98879. [7] E. Rouy and A. Tourin, A viscosity solutions approach to shape-from-shading, SIAM J. Num. Anal., 29 (1992), 867-884. doi: 10.1137/0729053. [8] J. Sethian, A fast marching level set method for monotonically advancing fronts, Proc. Nat. Acad. Sci., 93 (1996), 1591-1595. doi: 10.1073/pnas.93.4.1591. [9] J. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999. [10] H. Zhao, A fast sweeping method for eikonal equations, Math. Comput., 74 (2005), 603-627. doi: 10.1090/S0025-5718-04-01678-3. [11] H. Zhao, T. Chan, B. Merriman and S. Osher, A variational level set approach to multiphase motion, J. Comput. Phys., 127 (1996), 179-195. doi: 10.1006/jcph.1996.0167.
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