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

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

Published online: 2021-08

Preview Full PDF 141 3764
Export citation

Cited by

• 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.

65M12, 65M70, 65Z05

• BibTex
• RIS
• TXT
@Article{CSIAM-AM-2-460, author = {Chen , Wenbin and Wang , Shufen and Wang , Xiaoming}, 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
Copy to clipboard
The citation has been copied to your clipboard