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This work discusses the development, verification and performance assessment of a discontinuous Galerkin solver for the compressible Navier-Stokes equations using the Legion programming system. This is motivated by (i) the potential of this family of high-order numer- ical methods to accurately and efficiently realize scale-resolving simulations on unstructured grids and (ii) the desire to accommodate the utilization of emerging compute platforms that exhibit increased parallelism and heterogeneity. As a task-based programming model specifically designed for performance portability across distributed heterogeneous architectures, Legion represents an interesting lternative to the traditional approach of using Message Passing Interface for massively parallel computational physics solvers. Following detailed discussion of the implementation, the high-order convergence of the solver is demonstrated by a suite of canonical test cases and good strong scaling behavior is obtained. This work constitutes a first step towards a research platform that is able to be deployed and efficiently run on modern supercomputers.