Distance function and extension in normal direction for implicitly defined interfaces

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

Distance function and extension in normal direction for implicitly defined interfaces

Peter Frolkovič ^1, , Karol Mikula ^1, and Jozef Urbán ^1,

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

Abstract
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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].

Keywords: extrapolation, Distance function, fast marching method..

Mathematics Subject Classification: Primary: 35L50, 76M12; Secondary: 65L2.

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

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