Moment methods from the kinetic theory of gases exist as an alternative to the Navier-Stokes model. Models in this family are described by first-order hyperbolic PDEs with local relaxation. They provide a natural treatment for non-equilibrium effects and expand the regime for which the model is physically applicable past the Navier-Stokes level (when the continuum assumption breaks down). Discontinuous-Galerkin (DG) methods are very well suited for distributed parallel solution of first-order PDEs. This is because the optimal locality of the method minimizes needed communication between computational processes. One highly efficient, coupled space-time DG method that achieves third-order accuracy in both space and time while using only linear elements is the discontinuous-Galerkin Hancock (DGH) scheme, which was specifically designed for the efficient solution of PDEs resulting from moment closures. Third-order accuracy is obtained through the use of a technique originally proposed by Hancock. The combination of moment methods with the DGH discretization leads to a very efficient numerical treatment for viscous compressible gas flows that is accurate both in and out of local thermodynamic equilibrium. This thesis describe the first-ever implementation of this scheme for the solution of moment equations on large-scale distributed-memory computers. This implementation uses solution-directed automatic mesh refinement to increase accuracy while reducing cost. A linear hyperbolic-relaxation equation is used to verify the order of accuracy of the scheme. Next a supersonic compressible Euler case is used to demonstrate the mesh refinement as well as the scheme’s ability to capture sharp discontinuities. Third, a moment-closure is then used to compute a viscous mixing layer. This serves to demonstrate the ability of the first-order PDEs and the DG scheme to efficiently compute viscous solutions. A moment-closure is used to compute the solution for Stokes flow past a circular cylinder. This case reinforces the hyperbolic PDEs’ ability to accurately predict viscous phenomena. As this case is very low speed, it also demonstrates the numerical technique’s ability to accurately solve problems that are ill-conditioned due to the extremely low Mach number. Finally, the parallel efficiency of the scheme is evaluated on Canada’s largest supercomputer. It may be surprising to some that viscous flow behaviour can be accurately predicted by first-order PDEs. However, the applicability of hyperbolic moment methods to both continuum and non-equilibrium gas flows is now well established. Such a first-order treatment brings many physical and computational advantages to gas flow prediction.