A Level Set Algorithm for Tracking Discontinuities in Hyperbolic Conservation Laws II: Systems of Equations

A level set algorithm for tracking discontinuities in hyperbolic con-servation laws is presented. The algorithm uses a simple flnite difier-ence approach, analogous to the method of lines scheme presented in [20]. 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 the discontinuity. Also, 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 " of [6]. High order pointwise convergence is demonstrated for linear and nonlinear conservation laws, even at discontinuities and in multiple dimensions. The solutions are compared to standard high order shock capturing schemes. This paper focuses on scalar conser-vation laws. Level set tracking for systems of conservation laws in multi-dimensions will be presented in future work [2]