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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

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

[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

show all references

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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