High Order Numerical Discretization for Hamilton–Jacobi Equations on Triangular Meshes

Abstract

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 l2r which is improved here adjusting the dissipation term. The resulting first order scheme is e-monotonic (we explain the expression in the paper) and converges to the viscosity solution as {\cal O}(\sqrt{\Delta t}) 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.