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

We present unconditionally energy stable Runge-Kutta (RK) discontinuous Galerkin (DG) schemes for solving a class of fourth order gradient flows. Our algorithm is geared toward arbitrarily high order approximations in both space and time, while energy dissipation remains preserved without imposing any restriction 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 [J Sci. Comput. 77:467--501, 2018] for spatial discretization, we reformulate an extended linearized ODE system by the energy quadratization (EQ) approach. Second, we apply an s-stage algebraically stable RK method for temporal discretization. The resulting fully discrete DG schemes are linear and unconditionally energy stable. In addition, we introduce a prediction-correction procedure to improve both the accuracy and stability of the scheme. We illustrate the effectiveness of the proposed schemes by numerical tests with benchmark problems.