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We investigate the numerical performance of a Discontinuous Galerkin (DG)hydrodynamics implementation when applied to the problem of driven, isothermalsupersonic turbulence. While the high-order element-based spectral approach ofDG is known to efficiently produce accurate results for smooth problems(exponential convergence with expansion order), physical discontinuities insolutions, like shocks, prove challenging and may significantly diminish DG'sapplicability to practical astrophysical applications. We consider whether DGis able to retain its accuracy and stability for highly supersonic turbulence,characterized by a network of shocks. We find that our new implementation,which regularizes shocks at sub-cell resolution with artificial viscosity,still performs well compared to standard second-order schemes for moderatelyhigh Mach number turbulence, provided we also employ an additional projectionof the primitive variables onto the polynomial basis to regularize theextrapolated values at cell interfaces. However, the accuracy advantage of DGdiminishes significantly in the highly supersonic regime. Nevertheless, inturbulence simulations with a wide dynamic range that start with supersonicMach numbers and can resolve the sonic point, the low numerical dissipation ofDG schemes still proves advantageous in the subsonic regime. Our results thussupport the practical applicability of DG schemes for demanding astrophysicalproblems that involve strong shocks and turbulence, such as star formation inthe interstellar medium. We also discuss the substantial computational cost ofDG when going to high order, which needs to be weighted against the resultingaccuracy gain. For problems containing shocks, this favours the use ofcomparatively low DG order.

Supersonic turbulence simulations with GPU-based high-order Discontinuous Galerkin hydrodynamics