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Open AccessFeature PaperArticle

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

1
Department of Mathematics, Kwangwoon University, Seoul 01897, Korea
2
Institute of Mathematical Sciences, Ewha Womans University, Seoul 03760, Korea
3
Department of Mathematics, Ewha Womans University, Seoul 03760, Korea
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(12), 1242; https://doi.org/10.3390/math7121242
Received: 18 November 2019 / Revised: 12 December 2019 / Accepted: 13 December 2019 / Published: 16 December 2019
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. View Full-Text
Keywords: multi-component Cahn–Hilliard system; constrained convex splitting; unconditional unique solvability; unconditional energy stability; high-order time accuracy multi-component Cahn–Hilliard system; constrained convex splitting; unconditional unique solvability; unconditional energy stability; high-order time accuracy
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MDPI and ACS Style

Lee, H.G.; Shin, J.; Lee, J.-Y. A High-Order Convex Splitting Method for a Non-Additive Cahn–Hilliard Energy Functional. Mathematics 2019, 7, 1242.

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