We develop numerical tools and apply them to solve the relativistic Yang-Mills-Higgs equations in a model where the $SU\left(N\right)$ symmetry is spontaneously broken to its center. In $SU\left(2\right)$ and $SU\left(3\right)$, we obtain the different field profiles for infinite and finite center strings, with end points at external monopole sources. Exploration of parameter space permits the detection of a region where the equations get Abelianized. Finally, a general parametrization of the color structure of $SU\left(2\right)$ fields leads us to a reference point where an Abelian-like Bogomol’nyi-Prasad-Sommereld (BPS) bound is reconciled with $N$-ality.