The aim of this paper is to study a numerical scheme for a generalized planar Ginzburg-Landau energy in a circular geometry. The numerical scheme is based on the use of a spectral-Galerkin method. The stability analysis and an error estimate are presented. It is shown that the scheme is unconditionally stable. Some numerical simulation results are obtained using the scheme with various sets of boundary data, including those under the form \(u(\theta)=\exp (id\theta)\), where the integer \(d\) denotes the topological degree of the solution. Results include the computation of bifurcations from pure bend or splay patterns to spiral patterns for \(d=1\), energy decay curves for \(d=1\), spectral accuracy plots for \(d=2\) and computations of metastable or unstable higher-energy solutions as well as the lowest energy ground state solutions for values of \(d\) ranging from two to five.