  In this work, we propose and investigate stable high-order collocation-type
discretisations of the discontinuous Galerkin method on equidistant and
scattered collocation points. We do so by incorporating the concept of discrete
least squares into the discontinuous Galerkin framework. Discrete least squares
approximations allow us to construct stable and high-order accurate
approximations on arbitrary collocation points, while discrete least squares
quadrature rules allow us their stable and exact numerical integration. Both
methods are computed efficiently by using bases of discrete orthogonal
polynomials. Thus, the proposed discretisation generalises known classes of
discretisations of the discontinuous Galerkin method, such as the discontinuous
Galerkin collocation spectral element method. We are able to prove conservation
and linear $L^2$-stability of the proposed discretisations. Finally, numerical
tests investigate their accuracy and demonstrate their extension to nonlinear
conservation laws, systems, longtime simulations, and a variable coefficient
problem in two space dimensions.
