This work discusses a way of allowing fast implicit update schemes for the temporal discretization of phase-field models for gradient flow problems that employ Fourier-spectral methods for their spatial discretization. Through the repeated application of the Sherman–Morrison formula we provide a rule for approximations of the inverted tangent matrix of the corresponding Newton–Raphson method with a selectable order. Since the representation of this inversion is exact for a sufficiently high approximation order, the proposed scheme is shown to provide a fixed-point-type iterative solver for gradient flow problems that require the solution of linear systems in the context of an implicit time-integration. While the proposed scheme is applicable to general gradient flow phase-field models, we discuss the scheme in the context of the Cahn–Hilliard equation, the Swift–Hohenberg equation, and the phase-field crystal equation for which we demonstrate the performance of the proposed method in comparison with classical solvers.
%0 Journal Article
%1 KRISCHOK2024117220
%A Krischok, A.
%A Yaraguntappa, B.
%A Keip, M.-A.
%D 2024
%J Computer Methods in Applied Mechanics and Engineering
%K EXC2075 PN3 PN3-5(II) selected
%P 117220
%R https://doi.org/10.1016/j.cma.2024.117220
%T Fast implicit update schemes for Cahn–Hilliard-type gradient flow in the context of Fourier-spectral methods
%U https://www.sciencedirect.com/science/article/pii/S0045782524004766
%V 431
%X This work discusses a way of allowing fast implicit update schemes for the temporal discretization of phase-field models for gradient flow problems that employ Fourier-spectral methods for their spatial discretization. Through the repeated application of the Sherman–Morrison formula we provide a rule for approximations of the inverted tangent matrix of the corresponding Newton–Raphson method with a selectable order. Since the representation of this inversion is exact for a sufficiently high approximation order, the proposed scheme is shown to provide a fixed-point-type iterative solver for gradient flow problems that require the solution of linear systems in the context of an implicit time-integration. While the proposed scheme is applicable to general gradient flow phase-field models, we discuss the scheme in the context of the Cahn–Hilliard equation, the Swift–Hohenberg equation, and the phase-field crystal equation for which we demonstrate the performance of the proposed method in comparison with classical solvers.
@article{KRISCHOK2024117220,
abstract = {This work discusses a way of allowing fast implicit update schemes for the temporal discretization of phase-field models for gradient flow problems that employ Fourier-spectral methods for their spatial discretization. Through the repeated application of the Sherman–Morrison formula we provide a rule for approximations of the inverted tangent matrix of the corresponding Newton–Raphson method with a selectable order. Since the representation of this inversion is exact for a sufficiently high approximation order, the proposed scheme is shown to provide a fixed-point-type iterative solver for gradient flow problems that require the solution of linear systems in the context of an implicit time-integration. While the proposed scheme is applicable to general gradient flow phase-field models, we discuss the scheme in the context of the Cahn–Hilliard equation, the Swift–Hohenberg equation, and the phase-field crystal equation for which we demonstrate the performance of the proposed method in comparison with classical solvers.},
added-at = {2025-02-14T11:14:18.000+0100},
author = {Krischok, A. and Yaraguntappa, B. and Keip, M.-A.},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/277deaa29d33a00ec0bbbfdc6a7821220/simtechpuma},
doi = {https://doi.org/10.1016/j.cma.2024.117220},
interhash = {9cdc6ae973b077fcf415fa879f17cc04},
intrahash = {77deaa29d33a00ec0bbbfdc6a7821220},
issn = {0045-7825},
journal = {Computer Methods in Applied Mechanics and Engineering},
keywords = {EXC2075 PN3 PN3-5(II) selected},
pages = 117220,
timestamp = {2025-02-14T11:14:18.000+0100},
title = {Fast implicit update schemes for Cahn–Hilliard-type gradient flow in the context of Fourier-spectral methods},
url = {https://www.sciencedirect.com/science/article/pii/S0045782524004766},
volume = 431,
year = 2024
}
