Hamilton–Jacobi equation-based algorithms for approximate solutions to certain problems in applied geometry

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.