The half-planes problem for the level set equation

Abstract

The paper is dedicated to the construction of an analytic solution for the level set equation in R2 with an initial condition constituted by two half-planes. Such a problem can be seen as an equivalent Riemann problem in the Hamilton-Jacobi equation context. We rst rewrite the level set equation as a non-strictly hyperbolic problem and obtain a Riemann problem where the line sharing the initial discontinuity corresponds to the half-planes junction. Three dierent solutions corresponding to a shock, a rarefaction and a contact discontinuity are given in function of the two halfplanes con guration and we derive the solution for the level set equation. The study provides theoretical examples to test the numerical methods approaching the solution of viscosity of the level set equation. We perform simulations to check the three situations using a classical numerical method on a structured grid.