Energy Stable Arbitrary Order ETD-MS Method for Gradient Flows with Lipschitz Nonlinearity
Year: 2021
Author: Wenbin Chen, Shufen Wang, Xiaoming Wang
CSIAM Transactions on Applied Mathematics, Vol. 2 (2021), Iss. 3 : pp. 460–483
Abstract
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 kth order in time efficient linear numerical scheme with the help of an artificial regularization term of the form Aτk∂∂tLp(k)u where 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 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/csiam-am.2020-0033
CSIAM Transactions on Applied Mathematics, Vol. 2 (2021), Iss. 3 : pp. 460–483
Published online: 2021-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: Gradient flow epitaxial thin film growth exponential time differencing long time energy stability arbitrary order scheme multi-step method.
Author Details
Wenbin Chen Email
Shufen Wang Email
Xiaoming Wang Email
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