Application of the "Generalized Riemann Problem" method to 1-D compressible flows with material interfaces

The "Generalized Riemann Problem" (GRP) method is applied to 1-D compressible flows with material interfaces and variable cross section. The resulting scheme is second-order and uses a "mixed-type" grid, where cell boundaries can be either Lagrangian or Eulerian. In fact, using the analytic resolution of discontinuities at cell boundaries, provided by the GRP solution, one can extend the scheme presented here to include any adaptive mesh. Two numerical examples are studied: a planar shock-tube and exploding helium sphere. It is shown that discontinuities are sharply resolved while there are no oscillations in the smooth part of the flow. In particular, wave interactions, including formation of new shocks and reflection from the center of symmetry, are automatically taken care of.