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The idea of using fast sweeping methods for solving stationary systems ofconservation laws has previously been proposed for efficiently computingsolutions with sharp shocks. We further develop these methods to allow for amore challenging class of problems including problems with sonic points, shocksoriginating in the interior of the domain, rarefaction waves, andtwo-dimensional systems. We show that fast sweeping methods can producehigher-order accuracy. Computational results validate the claims of accuracy,sharp shock curves, and optimal computational efficiency.

Fast Sweeping Methods for Hyperbolic Systems of Conservation Laws at Steady State II