  In this paper, semi-discrete numerical scheme for the approximation of the
periodic Vlasov-viscous Burgers' system is developed and analyzed. The scheme
is based on the coupling of discontinuous Galerkin approximations for the
Vlasov equation and local discontinuous Galerkin approximations for the viscous
Burgers' equation. Both these methods use generalized numerical fluxes. The
proposed scheme is both mass and momentum conservative. Based on generalized
Gauss-Radau projections, optimal rates of convergence in the case of smooth
compactly supported initial data are derived. Finally, computational results
confirm our theoretical findings.
