The discontinuous Galerkin method with diffusion

We propose a way of extending the discontinuous Galerkin method from pure hyperbolic equations to convection-dominated equations with an $O(h)$ diffusion term. The resulting method is explicit and can be applied with polynomials of degree n ≥ 1. The extended method satisfies the same O(hn + 1/2) error estimate previously established for the discontinuous Galerkin method as applied to hyperbolic problems. Numerical results are provided.