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Efficient second-order, linear, decoupled and unconditionally energy stable schemes of the Cahn-Hilliard-Darcy equations for the Hele-Shaw flow

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Abstract

In this paper, we consider the numerical approximations for a hydrodynamical model of Cahn-Hilliard-Darcy equations. We develop two linear, decoupled, energy stable, and second-order time-marching schemes based on the “Invariant Energy Quadratization” method for nonlinear terms in the Cahn-Hilliard equation, and the projection method for the Darcy equations. Moreover, we prove the well-posedness of the linear system and their unconditional energy stabilities rigorously. We also construct a linear, decoupled, energy stable, and second-order time marching scheme by using the “Scalar Auxiliary Variable” method. Various numerical tests are presented to illustrate the stability and the accuracy of the numerical schemes and simulate the process of coarsening in binary fluid and investigate the effect of the rotating and the gravity on the Hele-Shaw cell in 2D and 3D.

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Funding

The work of R. Chen was partially supported by the National Natural Science Foundation of China (NSFC) with grant numbers 12001055 and 11971072. The work of K. Pan was partially supported by Science Challenge Project (No. TZ2016002), the National Natural Science Foundation of China (No. 41874086), the Excellent Youth Foundation of Hunan Province of China (No. 2018JJ1042). The work of X. Yang was partially supported by National Science Foundation with grant numbers DMS-1818783 and DMS-2012490.

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Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People’s Republic of China

    Rui Chen

  2. Department of Mathematics and Statistics, Hunan First Normal University, Changsha, Hunan, 410205, People’s Republic of China

    Yaxiang Li

  3. School of Mathematics and Statistics, Central South University, Changsha, 410083, People’s Republic of China

    Kejia Pan

  4. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Xiaofeng Yang

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  1. Rui Chen
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Correspondence to Yaxiang Li or Xiaofeng Yang.

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Chen, R., Li, Y., Pan, K. et al. Efficient second-order, linear, decoupled and unconditionally energy stable schemes of the Cahn-Hilliard-Darcy equations for the Hele-Shaw flow. Numer Algor 92, 2275–2306 (2023). https://doi.org/10.1007/s11075-022-01388-7

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