@Article{CSIAM-AM-2-3,
author = {Wenbin, Chen and Wang, Shufen and Xiaoming, Wang},
title = {Energy Stable Arbitrary Order ETD-MS Method for Gradient Flows with Lipschitz Nonlinearity},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2021},
volume = {2},
number = {3},
pages = {460--483},
abstract = {<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>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.2020-0033},
url = {https://global-sci.com/article/82353/energy-stable-arbitrary-order-etd-ms-method-for-gradient-flows-with-lipschitz-nonlinearity}
}