A numerical solution strategy for a one-dimensional field dislocation mechanics (FDM) model using the Discontinuous Galerkin (DG) method is developed. The FDM model is capable of simulating the dynamics of discrete, nonsingular dislocations using a partial differential equation involving a conservation law for the Burgers vector content with constitutive input for nucleation and velocity. Modeling of individual dislocation lines with an equilibrium compact core structure in the context of this continuum elastoplastic framework requires a non-convex stored energy density. Permanent deformation and stress redistribution caused by the dissipative transport of dislocations is modeled using thermodynamics-based constitutive laws. A DG method is employed to discretize the evolution equation of dislocation density yielding high orders of accuracy when the solution is smooth. The trade-offs of using a high order explicit Runge-Kutta time stepping and an implicit-explicit scheme are discussed. The developed numerical scheme is used to simulate the transport of a single screw dislocation wall in the case of a non-zero applied strain.