TY - JOUR
T1 - Fast High Order and Energy Dissipative Schemes with Variable Time Steps for Time-Fractional Molecular Beam Epitaxial Growth Model
AU - Hou , Dianming
AU - Qiao , Zhonghua
AU - Tang , Tao
JO - Annals of Applied Mathematics
VL - 4
SP - 429
EP - 461
PY - 2023
DA - 2023/11
SN - 39
DO - http://doi.org/10.4208/aam.OA-2023-0007
UR - https://global-sci.org/intro/article_detail/aam/22086.html
KW - Time-fractional molecular beam epitaxial growth, variable time-stepping
scheme, SAV approach, energy stability.
AB - <p style="text-align: justify;">In this paper, we propose and analyze high order energy dissipative time-stepping schemes for time-fractional molecular beam epitaxial (MBE)
growth model on the nonuniform mesh. More precisely, $(2−α)$-order, second-order and $(3−α)$-order time-stepping schemes are developed for the time-fractional MBE model based on the well known L1, L2-$1_σ,$ and L2 formulations
in discretization of the time-fractional derivative, which are all proved to be
unconditional energy dissipation in the sense of a modified discrete nonlocal-energy on the nonuniform mesh. In order to reduce the computational storage,
we apply the sum of exponential technique to approximate the history part of
the time-fractional derivative. Moreover, the scalar auxiliary variable (SAV)
approach is introduced to deal with the nonlinear potential function and the
history part of the fractional derivative. Furthermore, only first order method
is used to discretize the introduced SAV equation, which will not affect high&nbsp;order accuracy of the unknown thin film height function by using some proper
auxiliary variable functions $V (ξ)$. To our knowledge, it is the first time to unconditionally establish the discrete nonlocal-energy dissipation law for the modified
L1-, L2-$1_{σ^-},$ and L2-based high-order schemes on the nonuniform mesh, which
is essentially important for such time-fractional MBE models with low regular
solutions at initial time. Finally, a series of numerical experiments are carried
out to verify the accuracy and efficiency of the proposed schemes.</p>