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We present unconditionally energy stable Runge-Kutta (RK) discontinuousGalerkin (DG) schemes for solving a class of fourth order gradient flows. Ouralgorithm is geared toward arbitrarily high order approximations in both spaceand time, while energy dissipation remains preserved without imposing anyrestriction on time steps and meshes. We achieve this in two steps. First,taking advantage of the penalty free DG method introduced by Liu and Yin [JSci. Comput. 77:467--501, 2018] for spatial discretization, we reformulate anextended linearized ODE system by the energy quadratization (EQ) approach.Second, we apply an s-stage algebraically stable RK method for temporaldiscretization. The resulting fully discrete DG schemes are linear andunconditionally energy stable. In addition, we introduce aprediction-correction procedure to improve both the accuracy and stability ofthe scheme. We illustrate the effectiveness of the proposed schemes bynumerical tests with benchmark problems.

Energy stable Runge-Kutta discontinuous Galerkin schemes for fourth order gradient flows