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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 [1]. 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 [2]. High-order, point-wise convergence is demonstrated for linear and nonlinear scalar conservation laws, even at discontinuities and in multiple dimensions. The solutions are compared to standard high order shock capturing schemes.
This presentation will focus on systems of conservation laws. In particular, results of fully resolved detonation flows in the Euler equations will be presented. It will be demonstrated that the method can be used effectively when very accurate results are required for problems involving shock waves.
[1] C.-W. Shu and S. Osher, "Efficient Implementation of Essentially Non-oscillatory Shock-Capturing Schemes" Journal of Computational Physics, 77, 439-471, 1988.
[2] R. P. Fedkiw, T. Aslam, B. Merriman and S. Osher, "A Non-Oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (The Ghost Fluid Method)," Journal of Computational Physics, to appear, 152, 457-492, 1999.
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