TY - JOUR

T1 - On the stability and accuracy of partially and fully implicit schemes for phase field modeling

AU - Xu, Jinchao

AU - Li, Yukun

AU - Wu, Shuonan

AU - Bousquet, Arthur

N1 - Funding Information:
This work is supported in part by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials under contract number DE-SC0009249.

PY - 2019/3/1

Y1 - 2019/3/1

N2 - We study in this paper the accuracy and stability of partially and fully implicit schemes for phase field modeling. Through theoretical and numerical analysis of Allen–Cahn and Cahn–Hilliard models, we investigate the potential problems of using partially implicit schemes, demonstrate the importance of using fully implicit schemes and discuss the limitation of energy stability that are often used to evaluate the quality of a numerical scheme for phase-field modeling. In particular, we make the following observations: 1. a convex splitting scheme (CSS in short) can be equivalent to some fully implicit scheme (FIS in short) with a much different time scaling and thus it may lack numerical accuracy; 2. most implicit schemes are energy-stable if the time-step size is sufficiently small; 3. a traditionally known conditionally energy-stable scheme still possesses an unconditionally energy-stable physical solution; 4. an unconditionally energy-stable scheme is not necessarily better than a conditionally energy-stable scheme when the time step size is not small enough; 5. a first-order FIS for the Allen–Cahn model can be devised so that the maximum principle will be valid on the discrete level and hence the discrete phase variable satisfies |uh(x)|≤1 for all x and, furthermore, the linearized discretized system can be effectively preconditioned by discrete Poisson operators.

AB - We study in this paper the accuracy and stability of partially and fully implicit schemes for phase field modeling. Through theoretical and numerical analysis of Allen–Cahn and Cahn–Hilliard models, we investigate the potential problems of using partially implicit schemes, demonstrate the importance of using fully implicit schemes and discuss the limitation of energy stability that are often used to evaluate the quality of a numerical scheme for phase-field modeling. In particular, we make the following observations: 1. a convex splitting scheme (CSS in short) can be equivalent to some fully implicit scheme (FIS in short) with a much different time scaling and thus it may lack numerical accuracy; 2. most implicit schemes are energy-stable if the time-step size is sufficiently small; 3. a traditionally known conditionally energy-stable scheme still possesses an unconditionally energy-stable physical solution; 4. an unconditionally energy-stable scheme is not necessarily better than a conditionally energy-stable scheme when the time step size is not small enough; 5. a first-order FIS for the Allen–Cahn model can be devised so that the maximum principle will be valid on the discrete level and hence the discrete phase variable satisfies |uh(x)|≤1 for all x and, furthermore, the linearized discretized system can be effectively preconditioned by discrete Poisson operators.

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U2 - 10.1016/j.cma.2018.09.017

DO - 10.1016/j.cma.2018.09.017

M3 - Article

AN - SCOPUS:85058380042

VL - 345

SP - 826

EP - 853

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0374-2830

ER -