TY - JOUR
T1 - Energy Stable Arbitrary Order ETD-MS Method for Gradient Flows with Lipschitz Nonlinearity
AU - Chen , Wenbin
AU - Wang , Shufen
AU - Wang , Xiaoming
JO - CSIAM Transactions on Applied Mathematics
VL - 3
SP - 460
EP - 483
PY - 2021
DA - 2021/08
SN - 2
DO - http://doi.org/10.4208/csiam-am.2020-0033
UR - https://global-sci.org/intro/article_detail/csiam-am/19446.html
KW - Gradient flow, epitaxial thin film growth, exponential time differencing, long time energy stability, arbitrary order scheme, multi-step method.
AB - <p style="text-align: justify;">We present a methodology to construct efficient high-order in time accurate
numerical schemes for a class of gradient flows with appropriate Lipschitz continuous
nonlinearity. There are several ingredients to the strategy: the exponential time differencing (ETD), the multi-step (MS) methods, the idea of stabilization, and the technique
of interpolation. They are synthesized to develop a generic $k^{th}$ order in time efficient
linear numerical scheme with the help of an artificial regularization term of the form $Aτ^k\frac{∂}{∂t}\mathcal{L}^{p(k)}u$ where $\mathcal{L}$ is the positive definite linear part of the flow, $τ$ is the uniform
time step-size. The exponent $p(k)$ is determined explicitly by the strength of the Lipschitz nonlinear term in relation to $\mathcal{L}$ together with the desired temporal order of accuracy $k$. To validate our theoretical analysis, the thin film epitaxial growth without slope
selection model is examined with a fourth-order ETD-MS discretization in time and
Fourier pseudo-spectral in space discretization. Our numerical results on convergence
and energy stability are in accordance with our theoretical results.</p>