A fast-marching like algorithm for geometrical shock dynamics

We develop a new algorithm for the computation of the geometrical shock dynamics model (GSD). The method relies on the fast-marching paradigm and enables the discrete evaluation of the first arrival time of a shock wave and its local velocity on a cartesian grid. The proposed algorithm is based on a second order upwind finite difference scheme and reduces to a local nonlinear system of two equations solved by an iterative procedure. Reference solutions are built for a smooth radial configuration and for the 2D Riemann problem. The link between the GSD model and p-systems is given. Numerical experiments demonstrate the accuracy and the ability of the scheme to handle singularities.