  We investigate the numerical performance of a Discontinuous Galerkin (DG)
hydrodynamics implementation when applied to the problem of driven, isothermal
supersonic turbulence. While the high-order element-based spectral approach of
DG is known to efficiently produce accurate results for smooth problems
(exponential convergence with expansion order), physical discontinuities in
solutions, like shocks, prove challenging and may significantly diminish DG's
applicability to practical astrophysical applications. We consider whether DG
is 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 moderately
high Mach number turbulence, provided we also employ an additional projection
of the primitive variables onto the polynomial basis to regularize the
extrapolated values at cell interfaces. However, the accuracy advantage of DG
diminishes significantly in the highly supersonic regime. Nevertheless, in
turbulence simulations with a wide dynamic range that start with supersonic
Mach numbers and can resolve the sonic point, the low numerical dissipation of
DG schemes still proves advantageous in the subsonic regime. Our results thus
support the practical applicability of DG schemes for demanding astrophysical
problems that involve strong shocks and turbulence, such as star formation in
the interstellar medium. We also discuss the substantial computational cost of
DG when going to high order, which needs to be weighted against the resulting
accuracy gain. For problems containing shocks, this favours the use of
comparatively low DG order.
