A Level-Set Algorithm for Tracking Discontinuities in Hyperbolic Conservation Laws: I. Scalar Equations

A level-set algorithm for tracking discontinuities in hyperbolic conservation laws is presented. The algorithm uses a simple finite difference approach, analogous to the method of lines scheme presented in C.-W. Shu and S. Osher (1988, J. Comput. Phys.77, 439). 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 R. P. Fedkiw et al. (1999, J. Comput. phys.154, 459). 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 conservation laws. An example is given for shock tracking in the one-dimensional Euler equations. Level-set tracking for systems of conservation laws in multidimensions will be presented in future work.