East Asian J. Appl. Math., 13 (2023), pp. 886-913.
Published online: 2023-10
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Unconditionally stable and convergent variable-step BDF2 scheme for solving the MBE model with slope selection is derived. Discrete orthogonal convolution kernels of the variable-step BDF2 method are commonly utilized for solving the phase field models. We present new inequalities, concerning the vector forms, for the kernels especially dealing with nonlinear terms in the slope selection model. The convergence rate of the fully discrete scheme is proved to be two both in time and space in $L^2$ norm under the setting of the variable time steps. Energy dissipation law is proved rigorously with a modified energy by adding a small term to the discrete version of the original free energy functional. Two numerical examples including an adaptive time-stepping strategy are given to verify the convergence rate and the energy dissipation law.
}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.2022-286.271222}, url = {http://global-sci.org/intro/article_detail/eajam/22067.html} }Unconditionally stable and convergent variable-step BDF2 scheme for solving the MBE model with slope selection is derived. Discrete orthogonal convolution kernels of the variable-step BDF2 method are commonly utilized for solving the phase field models. We present new inequalities, concerning the vector forms, for the kernels especially dealing with nonlinear terms in the slope selection model. The convergence rate of the fully discrete scheme is proved to be two both in time and space in $L^2$ norm under the setting of the variable time steps. Energy dissipation law is proved rigorously with a modified energy by adding a small term to the discrete version of the original free energy functional. Two numerical examples including an adaptive time-stepping strategy are given to verify the convergence rate and the energy dissipation law.