Communications in Mathematical Sciences

Volume 21 (2023)

Number 1

Stabilized exponential time differencing schemes for the convective Allen–Cahn equation

Pages: 127 – 150

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n1.a6

Authors

Yongyong Cai (Laboratory of Mathematics and Complex Systems and School of Mathematical Sciences, Beijing Normal University, Beijing, China)

Lili Ju (Department of Mathematics, University of South Carolina, Columbia, S.C., U.S.A.)

Rihui Lan (Department of Mathematics, University of South Carolina, Columbia, S.C., U.S.A.)

Jingwei Li (Laboratory of Mathematics and Complex Systems and School of Mathematical Sciences, Beijing Normal University, Beijing, China)

Abstract

The convective Allen–Cahn equation has been widely used to simulate multi-phase flows in many phase-field models. As a generalized form of the classic Allen–Cahn equation, the convective Allen–Cahn equation still preserves the maximum bound principle (MBP) in the sense that the time-dependent solution of the equation with appropriate initial and boundary conditions preserves for all time a uniform pointwise bound in absolute value. In this paper, we develop efficient first- and second-order exponential time differencing (ETD) schemes combined with the linear stabilizing technique to preserve the MBP unconditionally in the discrete setting. The space discretization is done using the upwind difference scheme for the convective term and the central difference scheme for the diffusion term, and both the mobility and nonlinear terms are approximated through the linear convex interpolation. The unconditional preservation of the MBP of the proposed schemes is proven, and their convergence analysis is presented. Various numerical experiments in two and three dimensions are also carried out to verify the theoretical results.

Keywords

convective Allen–Cahn equation, maximum bound principle, Exponential time differencing, Stabilizing technique, Linear convex interpolation

2010 Mathematics Subject Classification

35B50, 35K55, 65M12

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Y. Cai’s work was partially supported by National Natural Science Foundation of China grants 12171041 and 11771036. L. Ju’s work was partially supported by US National Science Foundation grants DMS-1818438 and DMS-2109633. J. Li’s work was partially supported by China Postdoctoral Science Foundation grant 2021M700476.

Received 7 August 2021

Received revised 23 March 2022

Accepted 9 April 2022

Published 27 December 2022