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A High-Order Convex Splitting Method for a Non-Additive Cahn–Hilliard Energy Functional

Author

Listed:
  • Hyun Geun Lee

    (Department of Mathematics, Kwangwoon University, Seoul 01897, Korea)

  • Jaemin Shin

    (Institute of Mathematical Sciences, Ewha Womans University, Seoul 03760, Korea)

  • June-Yub Lee

    (Department of Mathematics, Ewha Womans University, Seoul 03760, Korea)

Abstract

Various Cahn–Hilliard (CH) energy functionals have been introduced to model phase separation in multi-component system. Mathematically consistent models have highly nonlinear terms linked together, thus it is not well-known how to split this type of energy. In this paper, we propose a new convex splitting and a constrained Convex Splitting (cCS) scheme based on the splitting. We show analytically that the cCS scheme is mass conserving and satisfies the partition of unity constraint at the next time level. It is uniquely solvable and energy stable. Furthermore, we combine the convex splitting with the specially designed implicit–explicit Runge–Kutta method to develop a high-order (up to third-order) cCS scheme for the multi-component CH system. We also show analytically that the high-order cCS scheme is unconditionally energy stable. Numerical experiments with ternary and quaternary systems are presented, demonstrating the accuracy, energy stability, and capability of the proposed high-order cCS scheme.

Suggested Citation

  • Hyun Geun Lee & Jaemin Shin & June-Yub Lee, 2019. "A High-Order Convex Splitting Method for a Non-Additive Cahn–Hilliard Energy Functional," Mathematics, MDPI, vol. 7(12), pages 1-13, December.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:12:p:1242-:d:298299
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    References listed on IDEAS

    as
    1. Lee, Hyun Geun & Kim, Junseok, 2008. "A second-order accurate non-linear difference scheme for the N -component Cahn–Hilliard system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(19), pages 4787-4799.
    2. Lee, Hyun Geun & Choi, Jeong-Whan & Kim, Junseok, 2012. "A practically unconditionally gradient stable scheme for the N-component Cahn–Hilliard system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(4), pages 1009-1019.
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