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research-article

Linearly implicit methods for Allen-Cahn equation

Authors:

Bülent KarasözenAuthors Info & Claims

Volume 450, Issue C

https://doi.org/10.1016/j.amc.2023.127984

Published: 01 August 2023 Publication History

Highlights

•

Construction of linearly implicit gradient stable methods by polarizing the energy functional.

•

A two-step version of the linearly implicit Kahan’s method provides accurate solutions with a less computational cost for Allen-Cahn equation.

•

Polarized and non-polarized energies dissipate at the same rate.

Abstract

It is well known that the Allen-Cahn equation satisfies a nonlinear stability property, i.e., the free-energy functional decreases in time. Linearly implicit integrators have been developed for energy-preserving methods for conservative systems with polynomial Hamiltonians, which are based on the concept of polarization. In this paper, we construct linearly implicit methods for gradient flows preserving the energy dissipation by polarizing the free-energy functional. Two-step linearly implicit methods are derived for the Allen-Cahn equation inheriting energy dissipation law. Numerical experiments for one-, two-, and three-dimensional Allen-Cahn equations demonstrate the energy dissipation and the accuracy of the linearly implicit methods.

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Cited By

Uzunca MKarasözen B(2025)Linearly implicit methods for the nonlinear Klein–Gordon equationMathematics and Computers in Simulation10.1016/j.matcom.2024.12.019231:C(318-330)Online publication date: 1-May-2025
https://dl.acm.org/doi/10.1016/j.matcom.2024.12.019

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Published In

cover image Applied Mathematics and Computation

Applied Mathematics and Computation Volume 450, Issue C

Aug 2023

369 pages

ISSN:0096-3003

Issue’s Table of Contents

Copyright © 2023.

Publisher

Elsevier Science Inc.

United States

Publication History

Published: 01 August 2023

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Cited By

Uzunca MKarasözen B(2025)Linearly implicit methods for the nonlinear Klein–Gordon equationMathematics and Computers in Simulation10.1016/j.matcom.2024.12.019231:C(318-330)Online publication date: 1-May-2025
https://dl.acm.org/doi/10.1016/j.matcom.2024.12.019

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