  A new evolution algorithm for the characteristic initial value problem based
upon an affine parameter rather than the areal radial coordinate used in the
Bondi-Sachs formulation is applied in the spherically symmetric case to the
gravitational collapse of a massless scalar field. The advantages over the
Bondi-Sachs version are discussed, with particular emphasis on the application
to critical collapse. Unexpected quadratures lead to a simple evolution
algorithm based upon ordinary differential equations which can be integrated
along the null rays. For collapse to a black hole in a Penrose compactified
spacetime, these equations are regularized throughout the exterior and interior
of the horizon up to the final singularity. They are implemented as a global
numerical evolution code based upon the Galerkin method. New results regarding
the global properties of critical collapse are presented.
