TY - JOUR
T1 - Errors of an Implicit Variable-Step BDF2 Method for a Molecular Beam Epitaxial Model with Slope Selection
AU - Zhao , Xuan
AU - Zhang , Haifeng
AU - Sun , Hong
JO - East Asian Journal on Applied Mathematics
VL - 4
SP - 886
EP - 913
PY - 2023
DA - 2023/10
SN - 13
DO - http://doi.org/10.4208/eajam.2022-286.271222
UR - https://global-sci.org/intro/article_detail/eajam/22067.html
KW - Molecular beam epitaxial growth, slope selection, variable-step BDF2 scheme, energy stability, convergence.
AB - <p style="text-align: justify;">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.</p>