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Volume 35, Issue 2
A Second-Order Length-Preserving and Unconditionally Energy Stable Rotational Discrete Gradient Method for Oseen-Frank Gradient Flows

Jie Xu, Xiaotian Yang & Zhiguo Yang

Commun. Comput. Phys., 35 (2024), pp. 369-394.

Published online: 2024-03

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  • Abstract

We present a second-order strictly length-preserving and unconditionally energy-stable rotational discrete gradient (Rdg) scheme for the numerical approximation of the Oseen-Frank gradient flows with anisotropic elastic energy functional. Two essential ingredients of the Rdg method are reformulation of the length constrained gradient flow into an unconstrained rotational form and discrete gradient discretization for the energy variation. Besides the well-known mean-value and Gonzalez discrete gradients, we propose a novel Oseen-Frank discrete gradient, specifically designed for the solution of Oseen-Frank gradient flow. We prove that the proposed Oseen-Frank discrete gradient satisfies the energy difference relation, thus the resultant Rdg scheme is energy stable. Numerical experiments demonstrate the efficiency and accuracy of the proposed Rdg method and its capability for providing reliable simulation results with highly disparate elastic coefficients.

  • AMS Subject Headings

65N35, 65N22, 65F05, 35J05

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COPYRIGHT: © Global Science Press

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@Article{CiCP-35-369, author = {Xu , JieYang , Xiaotian and Yang , Zhiguo}, title = {A Second-Order Length-Preserving and Unconditionally Energy Stable Rotational Discrete Gradient Method for Oseen-Frank Gradient Flows}, journal = {Communications in Computational Physics}, year = {2024}, volume = {35}, number = {2}, pages = {369--394}, abstract = {

We present a second-order strictly length-preserving and unconditionally energy-stable rotational discrete gradient (Rdg) scheme for the numerical approximation of the Oseen-Frank gradient flows with anisotropic elastic energy functional. Two essential ingredients of the Rdg method are reformulation of the length constrained gradient flow into an unconstrained rotational form and discrete gradient discretization for the energy variation. Besides the well-known mean-value and Gonzalez discrete gradients, we propose a novel Oseen-Frank discrete gradient, specifically designed for the solution of Oseen-Frank gradient flow. We prove that the proposed Oseen-Frank discrete gradient satisfies the energy difference relation, thus the resultant Rdg scheme is energy stable. Numerical experiments demonstrate the efficiency and accuracy of the proposed Rdg method and its capability for providing reliable simulation results with highly disparate elastic coefficients.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0191}, url = {http://global-sci.org/intro/article_detail/cicp/22944.html} }
TY - JOUR T1 - A Second-Order Length-Preserving and Unconditionally Energy Stable Rotational Discrete Gradient Method for Oseen-Frank Gradient Flows AU - Xu , Jie AU - Yang , Xiaotian AU - Yang , Zhiguo JO - Communications in Computational Physics VL - 2 SP - 369 EP - 394 PY - 2024 DA - 2024/03 SN - 35 DO - http://doi.org/10.4208/cicp.OA-2023-0191 UR - https://global-sci.org/intro/article_detail/cicp/22944.html KW - Nematic liquid crystal, Oseen-Frank gradient flow, energy stability, length-preservation, rotational discrete gradient method. AB -

We present a second-order strictly length-preserving and unconditionally energy-stable rotational discrete gradient (Rdg) scheme for the numerical approximation of the Oseen-Frank gradient flows with anisotropic elastic energy functional. Two essential ingredients of the Rdg method are reformulation of the length constrained gradient flow into an unconstrained rotational form and discrete gradient discretization for the energy variation. Besides the well-known mean-value and Gonzalez discrete gradients, we propose a novel Oseen-Frank discrete gradient, specifically designed for the solution of Oseen-Frank gradient flow. We prove that the proposed Oseen-Frank discrete gradient satisfies the energy difference relation, thus the resultant Rdg scheme is energy stable. Numerical experiments demonstrate the efficiency and accuracy of the proposed Rdg method and its capability for providing reliable simulation results with highly disparate elastic coefficients.

Jie Xu, Xiaotian Yang & Zhiguo Yang. (2024). A Second-Order Length-Preserving and Unconditionally Energy Stable Rotational Discrete Gradient Method for Oseen-Frank Gradient Flows. Communications in Computational Physics. 35 (2). 369-394. doi:10.4208/cicp.OA-2023-0191
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