Numerical Methods for Computing Discontinuous Solutions of aClass of Hamilton-Jacobi Equations Using aLevel Set Method

In this article, we first introduce aLax-Friedrichs type finite difference method to compute the $\mathrm{L}$-solution, following its original definition recently proposed by the second auther in[12] using level sets. We then generalize our numerical methods to compute the proper viscosity solution proposed in [11] for amore general class of HJ equations that includes conservation laws. We couple our numerical methods with asingular diffusive term of essential im-portance. With this singular viscosity, our numerical methods do not require the divergence structure of equations and do apply to more general equations developing shocks other than conservation laws. These numerical methods are generalized to higher order accuracy using WENO Local Lax-Friedrichs methods [17]. We verify that our numerical solutions approximate the proper viscosity solutions of [11] and, in particular, the entropy solutions in case of conservtion laws. 1Introductio