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An efficient space-time operator-splitting method for high-dimensional vector-valued Allen–Cahn equations

Yunxia Sun (College of Mathematics and System Sciences, Xinjiang University, Urumqi, China)
Xufeng Xiao (College of Mathematics and System Sciences, Xinjiang University, Urumqi, China)
Zhiming Gao (Institute of Applied Physics and Computational Mathematics, Beijing, China)
Xinlong Feng (College of Mathematics and System Sciences, Xinjiang University, Urumqi, China)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Article publication date: 19 June 2019

Issue publication date: 11 September 2019

Abstract

Purpose

The purpose of this paper is to propose an efficient space-time operator-splitting method for the high-dimensional vector-valued Allen–Cahn (AC) equations. The key of the space-time operator-splitting is to devide the complex partial differential equations into simple heat equations and nolinear ordinary differential equations.

Design/methodology/approach

Each component of high-dimensional heat equations is split into a series of one-dimensional heat equations in different spatial directions. The nonlinear ordinary differential equations are solved by a stabilized semi-implicit scheme to preserve the upper bound of the solution. The algorithm greatly reduces the computational complexity and storage requirement.

Findings

The theoretical analyses of stability in terms of upper bound preservation and mass conservation are shown. The numerical results of phase separation, evolution of the total free energy and total mass conservation show the effectiveness and accuracy of the space-time operator-splitting method.

Practical implications

Extensive 2D/3D numerical tests demonstrated the efficacy and accuracy of the proposed method.

Originality/value

The space-time operator-splitting method reduces the complexity of the problem and reduces the storage space by turning the high-dimensional problem into a series of 1D problems. We give the theoretical analyses of upper bound preservation and mass conservation for the proposed method.

Keywords

Acknowledgements

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped them to improve the results of this paper.

Citation

Sun, Y., Xiao, X., Gao, Z. and Feng, X. (2019), "An efficient space-time operator-splitting method for high-dimensional vector-valued Allen–Cahn equations", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 29 No. 9, pp. 3437-3453. https://doi.org/10.1108/HFF-01-2019-0076

Publisher

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Emerald Publishing Limited

Copyright © 2019, Emerald Publishing Limited