Flux Limiter for 2D Discontinuous Galerkin FEM I want to learn about implementing convection-diffusion simulations using discontinuous Galerkin (DG) finite element methods to solve $$ \dfrac{\partial c}{\partial t} = \nabla \cdot \mathbf{J}, $$ for species $c$ with diffusion coefficient $D$ within a fluid flowing with velocity $\mathbf{u}$ where $\mathbf{J} = D \nabla c - \mathbf{u} c$, such as in this example. The domain I'm solving this in has a sharp/shock discontinuity in it and my simulation is oscillating at this discontinuity. This behavior is expected, from what I've found, and it seems like a flux limiter would solve this issue and allow my solution to remain within the realm of realistic values. The part I'm working through with is how to actually implement the flux limiter in my simulation. How would would I implement such a correction (in 2D or 3D) in code using a DG finite element method? Any help would be welcome!

Check out Leveque's books or Hestaven's recent text. Both of these will help you understand limiters better.

They also provide code examples (online) for you to see how they implement limiters.