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An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

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Abstract

In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation, is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.

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Authors and Affiliations

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

    Xiao Li

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    ZhongHua Qiao

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

    Hui Zhang

Authors
  1. Xiao Li
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  2. ZhongHua Qiao
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  3. Hui Zhang
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Correspondence to ZhongHua Qiao.

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Li, X., Qiao, Z. & Zhang, H. An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation. Sci. China Math. 59, 1815–1834 (2016). https://doi.org/10.1007/s11425-016-5137-2

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  • DOI: https://doi.org/10.1007/s11425-016-5137-2

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