A New Discontinuous Galerkin Method for Hamilton-Jacobi Equations

In this paper we propose a new local discontinuous Galerkin method to directly solve Hamilton-Jacobi equations. The scheme is a natural extension of the monotone scheme. For the linear case, the method is equivalent to the discontinuous Galerkin method for conservation laws. Thus, stability and error analysis are obtained under the framework of conservation laws. For both convex and nonconvex Hamiltonian, optimal (k+1)-th order of accuracy for smooth solutions are obtained with piecewise k-th order polynomial approximations. The scheme is numerically tested on a variety of one and two dimensional problems. The method works well to capture sharp corners (discontinuous derivatives) and converges to the viscosity solution. AMS subject classification: 35Q53