Convergence of discontinuous Galerkin schemes for front propagation with obstacles

Olivier Bokanowski (1, 2) , Yingda Cheng (3) , Chi-Wang Shu (4)

1 LJLL - Laboratoire Jacques-Louis Lions
2 OC - Optimisation et commande
3 Department of Mathematics
4 Division of Applied Mathematics

We study semi-Lagrangian discontinuous Galerkin (SLDG) and Runge-Kutta discontinuous Galerkin (RKDG) schemes for some front propagation problems in the presence of an obstacle term, modeled by a nonlinear Hamilton-Jacobi equation of the form $\min(u_t + c u_x, u - g(x))=0$, in one space dimension. New convergence results and error bounds are obtained for Lipschitz regular data. These ``low regularity" assumptions are the natural ones for the solutions of the studied equations. Numerical tests are given to illustrate the behavior of our schemes.

Keywords: obstacle problems; dynamic programming principle; front propagation; discontinuous Galerkin methods; Hamilton-Jacobi-Bellman equations

Domains: Numerical Analysis [math.NA]

Olivier Bokanowski, Yingda Cheng, Chi-Wang Shu. Convergence of discontinuous Galerkin schemes for front propagation with obstacles. 2014. ⟨hal-00834342v3⟩