A mass conservative and energy stable scheme for the conservative Allen–Cahn type Ohta–Kawasaki model for diblock copolymers

Hyun Geun Lee; Hyun Geun Lee

doi:10.3934/math.2025307

AIMS Mathematics

2025, Volume 10, Issue 3: 6719-6731. doi: 10.3934/math.2025307

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A mass conservative and energy stable scheme for the conservative Allen–Cahn type Ohta–Kawasaki model for diblock copolymers

Hyun Geun Lee ^,

Department of Mathematics, Dongguk University, Seoul 04620, Republic of Korea

Received: 23 December 2024 Revised: 14 March 2025 Accepted: 17 March 2025 Published: 25 March 2025
MSC : 65M06, 65N06

The conservative Allen–Cahn type Ohta–Kawasaki model was introduced to reformulate the Cahn–Hilliard type Ohta–Kawasaki model. A difficulty in numerically solving the conservative Allen–Cahn type Ohta–Kawasaki model is how to discretize the nonlinear and nonlocal terms in time to preserve the mass conservation and energy decay properties without losing the efficiency and accuracy. To settle this problem, we present a linear, second-order, mass conservative, and energy stable scheme based on the Crank–Nicolson formula. In the scheme, the nonlinear and nonlocal terms are explicitly treated, which make the scheme linear, and the energy stability is guaranteed by adopting a truncated double-well potential and by adding two second-order stabilization terms. We analytically and numerically show that the scheme is mass conservative and energy stable. Additionally, the scheme can be easily implemented within a few lines of MATLAB code.
- conservative Allen–Cahn type Ohta–Kawasaki model,
- nonlocal Lagrange multiplier,
- diblock copolymers,
- mass conservation,
- energy stability
Citation: Hyun Geun Lee. A mass conservative and energy stable scheme for the conservative Allen–Cahn type Ohta–Kawasaki model for diblock copolymers[J]. AIMS Mathematics, 2025, 10(3): 6719-6731. doi: 10.3934/math.2025307

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Abstract

The conservative Allen–Cahn type Ohta–Kawasaki model was introduced to reformulate the Cahn–Hilliard type Ohta–Kawasaki model. A difficulty in numerically solving the conservative Allen–Cahn type Ohta–Kawasaki model is how to discretize the nonlinear and nonlocal terms in time to preserve the mass conservation and energy decay properties without losing the efficiency and accuracy. To settle this problem, we present a linear, second-order, mass conservative, and energy stable scheme based on the Crank–Nicolson formula. In the scheme, the nonlinear and nonlocal terms are explicitly treated, which make the scheme linear, and the energy stability is guaranteed by adopting a truncated double-well potential and by adding two second-order stabilization terms. We analytically and numerically show that the scheme is mass conservative and energy stable. Additionally, the scheme can be easily implemented within a few lines of MATLAB code.

References

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