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In this paper, we present error estimates of fully discrete Runge--Kuttadiscontinuous Galerkin (DG) schemes for linear time-dependent partialdifferential equations. The analysis applies to explicit Runge--Kutta timediscretizations of any order. For spatial discretization, a general discreteoperator is considered, which covers various DG methods, such as theupwind-biased DG method, the central DG method, the local DG method and theultra-weak DG method. We obtain error estimates for stable and consistent fullydiscrete schemes, if the solution is sufficiently smooth and a spatial operatorwith certain properties exists. Applications to schemes for hyperbolicconservation laws, the heat equation, the dispersive equation and the waveequation are discussed. In particular, we provide an alternative proof ofoptimal error estimates of local DG methods for equations with high orderderivatives in one dimension, which does not rely on energy inequalities ofauxiliary unknowns.

Error analysis of Runge--Kutta discontinuous Galerkin methods for linear time-dependent partial differential equations