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Numerical Approximations and Error Analysis of the Cahn–Hilliard Equation with Reaction Rate Dependent Dynamic Boundary Conditions

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Abstract

We consider numerical approximations and error analysis for the Cahn–Hilliard equation with reaction rate dependent dynamic boundary conditions (Knopf et al. ESAIM Math Model Numer Anal 55(1):229–282, 2021). Based on the stabilized linearly implicit approach, a first-order in time, linear and energy stable scheme for solving this model is proposed. The corresponding semi-discretized-in-time error estimates for the scheme are also derived. Numerical experiments, including the simulations with different energy potentials, the comparison with the former work, the convergence results for the relaxation parameter \(K\rightarrow 0\) and \(K\rightarrow \infty \) and the accuracy tests with respect to the time step size, are performed to validate the accuracy of the proposed scheme and the error analysis.

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Acknowledgements

The authors would like to thank Prof. Chun Liu for some useful discussions on the subject of this article. X. Bao is thankful to Prof. Chun Liu, Prof. Yiwei Wang, Prof. Qing Cheng and Prof. Tengfei Zhang for some stimulating discussions during the visit of Illinois Institute of Technology. X. Bao is also grateful to the Department of Applied Mathematics of Illinois Institute of Technology for the hospitality. X. Bao is partially supported by China Scholarship Council (No. 201906040019). H. Zhang is partially supported by the National Natural Science Foundation of China (Nos. 11971002 and 11471046).

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  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Xuelian Bao

  2. Laboratory of Mathematics and Complex Systems, Ministry of Education and School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Hui Zhang

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  1. Xuelian Bao
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  2. Hui Zhang
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Bao, X., Zhang, H. Numerical Approximations and Error Analysis of the Cahn–Hilliard Equation with Reaction Rate Dependent Dynamic Boundary Conditions. J Sci Comput 87, 72 (2021). https://doi.org/10.1007/s10915-021-01475-2

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  • DOI: https://doi.org/10.1007/s10915-021-01475-2

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