The hybrid method combining finite and boundary elements is presented and demonstrated to be suitable for analysis of the transient behavior of coaxial electromagnetic launcher, with the corresponding initial/boundary value problem formulated in terms of the hybrid potentials which mix the vector and scalar magnetic potentials. However, the numerical procedures for solution of the problem appear to be complicated, due to the coupling of these two methods. In this paper, the numerical details of the problem with axisymmetric geometry are studied. A number of numerical treatments and computational techniques are used to solve the problem efficiently and economically.