Five-dimensional gyrokinetic continuum simulations of electrostatic plasma turbulence in a straight, open-field-line geometry have been performed using a full-
discontinuous-Galerkin approach implemented in the Gkeyll code. While various simplifications have been used for now, such as long-wavelength approximations in the gyrokinetic Poisson equation and the Hamiltonian, these simulations include the basic elements of a fusion-device scrape-off layer: localised sources to model plasma outflow from the core, cross-field turbulent transport, parallel flow along magnetic field lines, and parallel losses at the limiter or divertor with sheath-model boundary conditions. The set of sheath-model boundary conditions used in the model allows currents to flow through the walls. In addition to details of the numerical approach, results from numerical simulations of turbulence in the Large Plasma Device, a linear device featuring straight magnetic field lines, are presented.