A level set algorithm for tracking discontinuities in hyperbolic conservation laws is presented in this paper. The algorithm uses a simple finite difference approach, analogous to a method of lines scheme. The zero of a level set function is used to specify the location of the discontinuity. Since a level set function is used to describe the front location, no extra data structures are needed to keep track of the location of discontinuity. Two solution states are used at all computational nodes, one corresponding to the real state and one corresponding to a ghost node state, analogous to the ghost fluid method. High-order pointwise convergence is demonstrated for scalar linear and nonlinear conservation laws, even at discontinuities and in multiple dimensions. The solutions here are compared to standard high order shock capturing schemes.

The paper focuses on the issues involved in tracking discontinuities in systems of conservation laws. Examples are presented of tracking contacts and hydrodynamic shocks in inert and chemically reacting compressible flow.