In this paper, we present a method to solve numerically the axisymmetric time-dependent Maxwell equations in a singular domain. In [Math. Methods Appl. Sci. 25 (2002) 49; Math. Methods Appl. Sci. 26 (2003) 861], the mathematical tools and an in-depth study of the problems posed in the meridian half-plane were exposed. The numerical method and experiments based on this theory are now described here. It is also the generalization to axisymmetric problems of the Singular Complement Method that we developed to solve Maxwell equations in 2D singular domains (see [C. R. Acad. Sci. Paris, t. 330 (2000) 391]). It is based on a splitting of the space of solutions in a regular subspace, and a singular one, derived from the singular solutions of the Laplace problem. Numerical examples are finally given, to illustrate our purpose. In particular, they show how the Singular Complement Method captures the singular part of the solution.