  In this paper, we present error estimates of fully discrete Runge--Kutta
discontinuous Galerkin (DG) schemes for linear time-dependent partial
differential equations. The analysis applies to explicit Runge--Kutta time
discretizations of any order. For spatial discretization, a general discrete
operator is considered, which covers various DG methods, such as the
upwind-biased DG method, the central DG method, the local DG method and the
ultra-weak DG method. We obtain error estimates for stable and consistent fully
discrete schemes, if the solution is sufficiently smooth and a spatial operator
with certain properties exists. Applications to schemes for hyperbolic
conservation laws, the heat equation, the dispersive equation and the wave
equation are discussed. In particular, we provide an alternative proof of
optimal error estimates of local DG methods for equations with high order
derivatives in one dimension, which does not rely on energy inequalities of
auxiliary unknowns.
