The half-planes problem for the level set equation

Stéphane Clain (1) , Malcom Djeno Ngomanda (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LMBP - Laboratoire de Mathématiques Blaise Pascal

The paper is dedicated to the construction of an analytic solution for the level set equation in $\mathbb R^2$ 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 first 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 different solutions corresponding to a shock, a rarefaction and a contact discontinuity are given in function of the two half-planes configuration 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.

Mots clés: Riemann problem; Cauchy problem; level set equation; analytical solution; half-planes problem

Domaines: Analyse numérique [math.NA]

Stéphane Clain, Malcom Djeno Ngomanda. The half-planes problem for the level set equation. International Journal of Numerical Analysis and Modeling, 2013, 10 (1), pp.99. ⟨hal-00467651⟩