Regularized linear schemes for the molecular beam epitaxy model with slope selection

In this paper, we propose full discrete linear schemes for the molecular beam epitaxy (MBE) model with slope selection, which are shown to be unconditionally energy stable and unique solvable. In details, using the invariant energy quadratization (IEQ) approach, along with a regularized technique, the MBE model is first discretized in time using either Crank-Nicolson or Adam-Bashforth strategies. The semi-discrete schemes are shown to be energy stable and unique solvable. Then we further use Fourier-spectral methods to discretize the space, ending with full discrete schemes that are energy-stable and unique solvable. In particular, the full discrete schemes are linear such that only a linear algebra problem need to be solved at each time step. Through numerical tests, we have shown a proper choice of the regularization parameter provides better stability and accuracy, such that larger time step is feasible. Afterward, we present several numerical simulations to demonstrate the accuracy and efficiency of our newly proposed schemes. The linearizing and regularizing strategy used in this paper could be readily applied to solve a class of phase field models that are derived from energy variation.