High accuracy modeling of sharp wave fronts for hyperbolic problems

In this paper, the arbitrary order derivative (ADER) schemes based on the generalized Riemann problem are proposed to capture shock waves and contact discontinuities by coupling ghost fluid method (GFM). The reconstruction technique for spatial derivatives at cell boundaries is presented by piece-wise smooth WENO interpolations which are used as initial states of the Riemann problems. A level set function is used to keep track of the location of wave fronts. The shock waves are pushed forward by shock speeds which are obtained by the Rankine-Hugoniot conditions, whereas the contact discontinuities are advanced by local fluid velocities. Numerical examples show that the presented scheme is suitable for capturing fine flow structures and has an accuracy comparable to other methods designed for traditional contact discontinuity.