Geodesic
Hamilton–Jacobi equation-based algorithms for approximate solutions to certain problems in applied geometry
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 8, pp. 1345-1358

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Two important applied geometry problems are solved numerically. One is that of determining the nearest boundary distance from an arbitrary point in a domain. The other is that of determining (in a shortest-path metric) the distance between two points with the obstacles boundaries traversed inside the domain. These problems are solved by the time relaxation method as applied to a nonlinear Hamilton–Jacobi equation. Two major approaches are taken. In one approach, an equation with elliptic operators on the right-hand side is derived by changing the variables in the eikonal equation with viscous terms. In the other approach, first- and second-order monotone Godunov schemes are constructed taking into account the hyperbolicity of the nonlinear eikonal equation. One- and two-dimensional problems are solved to demonstrate the performance of the developed numerical algorithms and to examine their properties. Application problems are solved as examples. 
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     title = {Hamilton{\textendash}Jacobi equation-based algorithms for approximate solutions to certain problems in applied geometry},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     publisher = {mathdoc},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_8_a1/}
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D. I. Ivanov; I. E. Ivanov; I. A. Kryukov. Hamilton–Jacobi equation-based algorithms for approximate solutions to certain problems in applied geometry. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 8, pp. 1345-1358. http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_8_a1/