Volume 2, Issue 3
Energy Stable Arbitrary Order ETD-MS Method for Gradient Flows with Lipschitz Nonlinearity

Wenbin Chen, Shufen Wang & Xiaoming Wang

CSIAM Trans. Appl. Math., 2 (2021), pp. 460-483.

Published online: 2021-08

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  • 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 $k^{th}$ 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.

  • Keywords

Gradient flow, epitaxial thin film growth, exponential time differencing, long time energy stability, arbitrary order scheme, multi-step method.

  • AMS Subject Headings

65M12, 65M70, 65Z05

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COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-2-460, author = {Wenbin and Chen and and 18067 and and Wenbin Chen and Shufen and Wang and and 18068 and and Shufen Wang and Xiaoming and Wang and and 18069 and 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 = {

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.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.2020-0033}, url = {http://global-sci.org/intro/article_detail/csiam-am/19446.html} }
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 -

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.

Wenbin Chen, Shufen Wang & XiaomingWang. (2021). Energy Stable Arbitrary Order ETD-MS Method for Gradient Flows with Lipschitz Nonlinearity. CSIAM Transactions on Applied Mathematics. 2 (3). 460-483. doi:10.4208/csiam-am.2020-0033
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