  An implicit, fully characteristic, numerical scheme for solving the field
equations of a cosmic string coupled to gravity is described. The inclusion of
null infinity as part of the numerical grid allows us to apply suitable
boundary conditions on the metric and matter fields to suppress unphysical
divergent solutions. The code is tested by comparing the results with exact
solutions, checking that static cosmic string initial data remain constant when
evolved and undertaking a time dependent convergence analysis of the code. It
is shown that the code is accurate, stable and exhibits clear second order
convergence. The code is used to analyse the interaction between a
Weber--Wheeler pulse of gravitational radiation with the string. The
interaction causes the string to oscillate at frequencies inversely
proportional to the masses of the scalar and vector fields of the string. After
the pulse has largely radiated away the string continues to ring but the
oscillations slowly decay and eventually the variables return to their static
values.
