High order numerical discretization for Hamilton-Jacobi equations on triangular meshes

In this paper we construct several numerical approximations for first order Hamilton-Jacobi equations on triangular meshes. We show that, thanks to a filtering procedure, the high order versions are non-oscillatory in the sense of satisfying the maximum principle. The methods are based on the first order Lax-Friedrichs scheme [2] which is improved here adjusting the dissipation term. The resulting first order scheme is ε-monotonic (we explain the expression in the paper) and converges to the viscosity solution as O(√Δt) for the L∞-norm. The first high order method is directly inspired by the ENO philosophy in the sense where we use the monotonic Lax-Friedrichs Hamiltonian to reconstruct our numerical solutions. The second high order method combines a spatial high order discretization with the classical high order Runge-Kutta algorithm for the time discretization. Numerical experiments are performed for general Hamiltonians and L1, L2 and L∞-errors with convergence rates calculated in one and two space dimensions show the k-th order rate when piecewise polynomial of degree k functions are used, measured in L1-norm.