Free evolution of self-gravitating, spherically symmetric waves

We perform a numerical free evolution of a self-gravitating, spherically symmetric scalar field satisfying the wave equation. The evolution equations can be written in a very simple form and are symmetric hyperbolic in the Eddington-Finkelstein coordinates. The simplicity of the system allows us to display and deal with the typical gauge instability present in these coordinates. The numerical evolution is performed with a standard method of lines, fourth order in space and time. The evolution is performed using the standard Runge-Kutta method while the space discrete derivative is symmetric (nondissipative). The constraints are preserved, within numerical errors, by the evolution and we are able to reproduce several known results.