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Linear numerical schemes for epitaxial thin film growth model with energy stability

A few linear schemes for nonlinear PDE model of thin film growth model without slope selection are presented in the talk. In the first order linear scheme, the idea of convex-concave decomposition of the energy functional is applied, and the particular decomposition places the nonlinear term in the concave part of the energy, in contrast to a standard convexity splitting scheme. As a result, the numerical scheme is fully linear at each time step and unconditionally solvable, and an unconditional energy stability is guaranteed by the convexity splitting nature of the numerical scheme. To improve the numerical accuracy, a second order temporal approximation for the nonlinear term is recently reported, which preserves an energy stability. To solve this highly non-trivial nonlinear system, a linear iteration algorithm is proposed, with an introduction of a second order artificial diffusion term. Moreover, a contraction mapping property is proved for such a linear iteration. Finally, a linear second order scheme is proposed and analyzed, so that the energy stability is assured at a theoretical level. Some numerical simulation results are also presented in the talk.