Numerical-analytic construction of a generalized solution to the eikonal equation in the plane case

Abstract

In the 1970s Kruzhkov introduced the concept of a generalized solution to the eikonal equation, and, for a medium of constant refractive index, indicated a class of functions containing the generalized solution to the Dirichlet boundary value problem. We present constructive methods for finding such a solution in the plane case. Nonsmooth singularities are known to develop in generalized solutions owing to the presence of pseudovertices, which are singular points of the boundary of the boundary set. Identifying such pseudovertices is related to finding fixed points of mappings formed in a local reparameterization of this boundary. We obtain necessary conditions for the existence of pseudovertices in the case when the curvature of the parametrically defined boundary of boundary sets is not smooth. These conditions are written as an equation for the pseudovertex marker (a numerical measure of the local nonconvexity of the boundary set). This equation, which has the typical structure of constructions involving fixed points, can be reduced to an algebraic equation. The solution of this equation (the marker) is found analytically if the curvature of the boundary of the boundary set has a nonsmooth extremum at a pseudovertex. We also give an example of a numerical-analytic construction of a generalized solution to the boundary value problem, with indication of the singular set and the evolution of wave fronts. Bibliography: 29 titles.