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Reduced-Order Modelling for the Allen-Cahn Equation Based on Scalar Auxiliary Variable Approaches

Reduced-Order Modelling for the Allen-Cahn Equation Based on Scalar Auxiliary Variable Approaches
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Year:    2019

Author:    Xiaolan Zhou, Mejdi Azaiez, Chuanju Xu

Journal of Mathematical Study, Vol. 52 (2019), Iss. 3 : pp. 258–276

Abstract

In this article, we study the reduced-order modelling for Allen-Cahn equation. First, a collection of phase field data, i.e., an ensemble of snapshots of at some time instances is obtained from numerical simulation using a time-space discretization. The full discretization makes use of a temporal scheme based on the scalar auxiliary variable approach and a spatial spectral Galerkin method. It is shown that the time stepping scheme is unconditionally stable. Then a reduced order method is developed using by proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM). It is well-known that the Allen-Cahn equations have a nonlinear stability property, i.e., the free-energy functional decreases with respect to time. Our numerical experiments show that the discretized Allen-Cahn system resulting from the POD-DEIM method inherits this favorable property by using the scalar auxiliary variable approach. A few numerical results are presented to illustrate the performance of the proposed reduced order method. In particular, the numerical results show that the computational efficiency is significantly enhanced as compared to directly solving the full order system.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jms.v52n3.19.03

Journal of Mathematical Study, Vol. 52 (2019), Iss. 3 : pp. 258–276

Published online:    2019-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    19

Keywords:    Allen-Cahn equation scalar auxiliary variable proper orthogonal decomposition discrete empirical interpolation method.

Author Details

Xiaolan Zhou

Mejdi Azaiez

Chuanju Xu

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