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Volume 11, Issue 2
An Energy Stable Second-Order Accurate Scheme for Microphase Separation of Periodic Diblock Copolymers

Junxiang Yang & Junseok Kim

DOI: 10.4208/eajam.240620.071020

East Asian J. Appl. Math., 11 (2021), pp. 234-254.

Published online: 2021-02

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  • Abstract

A linear, unconditionally energy stable, and second-order accurate numerical scheme for the Ohta-Kawasaki equation modeling the diblock copolymer dynamics is proposed. The temporal discretisation is based on the Crank-Nicolson temporal discretisation and extrapolation. To suppress the dominance of nonlinear term, a proper stabilising parameter is used. All nonlinear parts are linearised by using the extrapolation from the information at preceding time levels. To solve the resulting linear system, an efficient linear multigrid algorithm is used. The unconditionally energy stability, mass conservation, and unique solvability of the scheme are analytically proved. In two-dimensional case, we run convergence and stability tests, and consider pattern formations for various average concentrations. Pattern formations in three-dimensional space are also studied.

  • Keywords

Unconditional energy stability, second-order accuracy, Ohta-Kawasaki model, finite difference method.

  • AMS Subject Headings

35K35, 35Q35, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-11-234, author = {Yang , Junxiang and Kim , Junseok}, title = {An Energy Stable Second-Order Accurate Scheme for Microphase Separation of Periodic Diblock Copolymers}, journal = {East Asian Journal on Applied Mathematics}, year = {2021}, volume = {11}, number = {2}, pages = {234--254}, abstract = {

A linear, unconditionally energy stable, and second-order accurate numerical scheme for the Ohta-Kawasaki equation modeling the diblock copolymer dynamics is proposed. The temporal discretisation is based on the Crank-Nicolson temporal discretisation and extrapolation. To suppress the dominance of nonlinear term, a proper stabilising parameter is used. All nonlinear parts are linearised by using the extrapolation from the information at preceding time levels. To solve the resulting linear system, an efficient linear multigrid algorithm is used. The unconditionally energy stability, mass conservation, and unique solvability of the scheme are analytically proved. In two-dimensional case, we run convergence and stability tests, and consider pattern formations for various average concentrations. Pattern formations in three-dimensional space are also studied.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.240620.071020 }, url = {http://global-sci.org/intro/article_detail/eajam/18633.html} }
TY - JOUR T1 - An Energy Stable Second-Order Accurate Scheme for Microphase Separation of Periodic Diblock Copolymers AU - Yang , Junxiang AU - Kim , Junseok JO - East Asian Journal on Applied Mathematics VL - 2 SP - 234 EP - 254 PY - 2021 DA - 2021/02 SN - 11 DO - http://doi.org/10.4208/eajam.240620.071020 UR - https://global-sci.org/intro/article_detail/eajam/18633.html KW - Unconditional energy stability, second-order accuracy, Ohta-Kawasaki model, finite difference method. AB -

A linear, unconditionally energy stable, and second-order accurate numerical scheme for the Ohta-Kawasaki equation modeling the diblock copolymer dynamics is proposed. The temporal discretisation is based on the Crank-Nicolson temporal discretisation and extrapolation. To suppress the dominance of nonlinear term, a proper stabilising parameter is used. All nonlinear parts are linearised by using the extrapolation from the information at preceding time levels. To solve the resulting linear system, an efficient linear multigrid algorithm is used. The unconditionally energy stability, mass conservation, and unique solvability of the scheme are analytically proved. In two-dimensional case, we run convergence and stability tests, and consider pattern formations for various average concentrations. Pattern formations in three-dimensional space are also studied.

Yang , Junxiang and Kim , Junseok. (2021). An Energy Stable Second-Order Accurate Scheme for Microphase Separation of Periodic Diblock Copolymers. East Asian Journal on Applied Mathematics. 11 (2). 234-254. doi:10.4208/eajam.240620.071020
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