TY - JOUR
T1 - A high-order convex splitting method for a non-additive Cahn-Hilliard energy functional
AU - Lee, Hyun Geun
AU - Shin, Jaemin
AU - Lee, June Yub
N1 - Funding Information:
Funding: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Korea government MSIP (2017R1D1A1B0-3032422, 2017R1E1A1A0-3070161, and 2019R1C1C1011112).
Publisher Copyright:
© 2019 by the authors.
PY - 2019/12/1
Y1 - 2019/12/1
N2 - 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.
AB - 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.
KW - Constrained convex splitting
KW - High-order time accuracy
KW - Multi-component Cahn-Hilliard system
KW - Unconditional energy stability
KW - Unconditional unique solvability
UR - http://www.scopus.com/inward/record.url?scp=85079692300&partnerID=8YFLogxK
U2 - 10.3390/MATH7121242
DO - 10.3390/MATH7121242
M3 - Article
AN - SCOPUS:85079692300
SN - 2227-7390
VL - 7
JO - Mathematics
JF - Mathematics
IS - 12
M1 - 1242
ER -