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Energy Stable Arbitrary Order ETD-MS Method for Gradient Flows with Lipschitz Nonlinearity

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 AτktLp(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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